UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


GIFT  OF 

I-.IISS   BETTY   JA1ISS 


WENTWORTH-SMITH  MATHEMATICAL  SERIES 


JUNIOR  HIGH  SCHOOL 
MATHEMATICS 

BOOK  I 

BY 

GEORGE  WENTWORTH 
DAVID  EUGENE  SMITH 

AND 

JOSEPH  CLIFTON  BROWN 


GINN  AND  COMPANY 

BOSTON     •    NEW   YORK    •    CHICAGO     •    LONDON 
ATLANTA    •    DALLAS     •    COLUMBUS     •    SAN   FRANCISCO 


COPYRIGHT,  1917,  BY  GEORGE   WENTWORTH 

DAVID  EUGENE  SMITH,  AXD  JOSEPH  CLIFTON  BROWN 

ENTERED  AT  STATIONERS'  HALL 

ALL  RIGHTS  RESERVED 

520.1 


tgfre  gtfttnacum 

CINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


V- 


PEEFACE 

A  proper  curriculum  for  junior  high  schools  and  six-year 
T—  high  schools  demands,  in  the  opinion  of  many  teachers,  a  course 
in>  in  mathematics  which  introduces  concrete,  intuitional  geometry 
c^  and  the  simple  uses  of  algebra  in  the  lower  classes.  This  book 

9=       is  intended  to  meet  such  a  demand  for  the  lowest  class. 
<c 

Arithmetic  furnishes  the  material  for  the  first  half  of  the 
book.  The  second  half  of  the  book  is  devoted  to  intuitional 
and  constructive  geometry,  a  subject  which  is  more  concrete 

uo         than  algebra  and  which  admits  of  more  simple  illustration. 

-5  Both  arithmetic  and  geometry  are  arranged  with  respect  to 
large  topics,  the  intention  being  to  avoid  the  lack  of  system 
which  so  often  deprives  the  student  of  that  feeling  of  mastery 

^  which  is  his  right  and  his  privilege.  These  large  topics  are 
set  forth  clearly  in  the  table  of  contents. 

In  this  book  there  is  gradually  introduced  the  algebraic 
formula,  so  that  the  student  is  aware  of  the  value  of  algebra 
as  a  working  tool  before  he  proceeds  to  the  study  of  Book  II. 
The  text  is  so  arranged  that  with  entire  convenience  to  the 
teacher  and  student  the  work  in  arithmetic  may  be  taken 
parallel  with  the  work  in  geometry,  or  the  work  in  geometry* 
may  precede  the  work  in  arithmetic. 

Any  remodeling  of  the  elementary  curriculum  that  sacrifices 
thorough  training  in  arithmetic  would  be  a  transitory  thing, 

ij 

and  any  anaemic  course  in  mathematics  that  leaves  a  student 
t  too  languid  intellectually  to  pursue  the  subject  further  with 
success  is  foredoomed  to  failure.  This  book  gives  to  arithmetic 
the  place  due  to  its  fundamental  importance,  and  it  adheres 
to  a  sane  and  usable  topical  plan  throughout  the  development 

iii 


238174 


iv  PEEFACE 

of  the  various  subjects  treated.  Because  of  this  the  authors 
believe  that  they  have  here  produced  a  textbook  suited  to  the 
needs  of  a  rapidly  growing  class  of  schools  and  that  they  have 
not  failed  in  any  respect  to  adhere  to  the  best  standards  of 
mathematics  and  pedagogy.  As  to  material  for  daily  drill, 
teachers  should  consult  page  105. 

Every  student  who  uses  this  book  will  find  it  convenient  to 
have  a  protractor  like  the  one  represented  on  page  115.  Ginn 
and  Company  are  prepared  to  furnish  such  protractors,  made 
of  transparent  celluloid. 

Book  II  is  devoted  to  algebra  and  arithmetic,  each  making 
use  of  the  important  facts  presented  in  Book  I  and  each 
including  those  large  and  important  topics  which  are  valuable 
in  the  elementary  education  of  every  boy  and  girl.  The  two 
books  thus  work  together  to  a  common  purpose,  the  first  being 
the  more  concrete  and  preparing  by  careful  steps  for  the 
second,  and  the  second  blending  with  the  first  in  presenting 
to  the  student  a  well-organized  foundation  for  the  more  formal 
treatment  of  the  mathematics  which  naturally  follows. 

The  authors  take  great  pleasure  in  being  able  to  include  in 
this  work  a  number  of  decorative  illustrations  of  early  mathe- 
matical instruments  and  their  uses,  by  Mr.  T.  M.  Cleland. 
They  feel  sure  that  teachers  and  students  will  welcome  this 
innovation  in  the  preparation  of  textbooks  in  mathematics, 
and  will  appreciate  such  a  combination  of  the  work  of  the 
artist  with  that  of  the  mathematician. 


CONTENTS 

PART  I.    ARITHMETIC 

CHAPTER  PAGE 

I.    ARITHMETIC  OF  THE  HOME 1 

II.    ARITHMETIC  OF  THE  STORE 33 

III.  ARITHMETIC  OF  THE  FARM 55 

IV.  ARITHMETIC  OF  INDUSTRY 67 

V.   ARITHMETIC  OF  THE  BANK 77 

MATERIAL  FOR  DAILY  DRILL 105 

PART  II.    GEOMETRY 

I.    GEOMETRY  OF  FORM .  Ill 

II.    GEOMETRY  OF  SIZE 155 

III.  GEOMETRY  OF  POSITION 215 

IV.  SUPPLEMENTARY  WORK 237 

TABLES  FOR  REFERENCE 247 

INDEX  249 


JUNIOR  HIGH  SCHOOL 
MATHEMATICS 

BOOK  I 

PART  I.   ARITHMETIC 
I.   ARITHMETIC  OF  THE  HOME 

Nature  of  the  Work.  You  have  already  completed  the 
arithmetic  which  treats  of  ordinary  computation,  and  have 
probably  learned  how  to  use  the  common  tables  of  meas- 
ures and  how  to  find  a  given  per  cent  of  a  number.  You  are 
therefore  now  ready  to  consider  the  most  important  appli- 
cations of  arithmetic  —  those  which  relate  to  the  home,  the 
store,  the  farm,  the  most  common  industries,  and  the  bank. 

In  taking  up  these  applications  of  arithmetic  we  shall 
review  the  operations  with  numbers  and  shall  pay  par- 
ticular attention  to  those  short  cuts  in  computation  that 
the  business  man  needs  to  know  and  that  are  useful  in 
various  kinds  of  work.  We  shall  also  take  up  again  the 
important  subject  of  percentage,  which  enters  into  every 
kind  of  business,  and  shall  treat  of  it  from  the  beginning. 

In  this  book  notes  in  this  type  are  chiefly  for  the  teacher's  use. 

The  teacher  is  advised  to  spend  a  little  time  in  discussing  this 
page,  in  order  to  take  stock  of  what  has  already  been  studied  and 
to  set  forth  the  general  nature  of  the  work  which  is  ahead  of  the 
students.  The  reading  of  a  few  interesting  problems  which  the 
students  will  meet  adds  a  motive  to  the  work. 

1 


ARITHMETIC  OF  THE  HOME 


Cash  Account.  It  is  important  that  every  boy  and  girl 
should  early  in  life  form  the  habit  of  keeping  a  personal 
cash  account.  It  is  customary  to  write  the  receipts  on  the 
left  side  of  the  account  (called  the  debit  side)  and  to  write 
the  payments  on  the  right  side  (called  the  credit  side),  thus : 

RECEIPTS  PAYMENTS 


iqso 

Qam,. 


276 

60 
20 


3_8_0_ 
~3~00 


jlam,. 


8  fiowk, 


06 

26 

50 

300 

~380 


Here  a  shows  cash  on  hand  when  this  page  of  the 
account  is  begun ;  5,  <?,  and  d  are  receipts ;  e  is  the  sum 
of  these  items  on  the  day  the  account  is  balanced.  On  the 
right  side  </,  A,  and  i  are  amounts  paid.  To  find  the  amount 
on  hand  we  subtract  80$,  the  sum  of  5$,  25$,  and  50$, 
from  $3.80,  and  find  that  the  balance  is  |3.00,  which  we 
write  at  j  and  again  at  f.  To  check  the  work,  g,  A,  i,  and  / 
are  added,  and  the  sums,  at  e  and  k,  must  agree.  In 
keeping  a  cash  account  the  symbol  $  is  usually  omitted. 

Check.  To  check  the  work  in  addition  always  add  a 
column  from  the  top  down  after  having  added  it  from 
the  bottom  up.  Every  computation  shoidd  be  checked. 

One  of  the  first  things  one  must  learn  in  business  is  the  neces- 
sity of  checking  every  computation.  Teachers  should  impress  this 
important  business  rule  upon  the  students.  When  the  other  opera- 
tions are  met  in  this  work,  the  teacher  should  explain  the  proper 
checks  if  necessary,  but  these  checks  should  be  known  to  the  students 
from  the  work  of  previous  years. 


CASH  ACCOUNTS  3 

Exercise  1.    Cash  Accounts 

Given  the  following  items,  make  out  the  cash  accounts  and 
balance  them : 

1.  Receipts:  Oct.  25,  cash  on  hand,  45$;  Oct.  26,  earned 
by  errands,  26$;  Oct.  27,  earned  by  errands,  35$.  Payments: 
Oct.  26,  ball,  20$;  Oct.  28,  bat,  25$.    Balanced  Oct.  29. 

2.  Receipts:    Sept.  7,    cash   on   hand,    |3.20 ;    Sept.  8, 
earned  by  errands,  40$;  Sept.  9,  gift  from  father,  50$; 
Sept.  10,  earned  by  cleaning  automobile,  30$.    Payments : 
Sept.  9,  football  pants,  $1.75;  Sept.  10,  share  in  football, 
50$;  Sept  11,  book,  30$.    Balanced  Sept.  12. 

3.  Receipts:  Nov.  5,  cash  on  hand,  75$;  Nov.  5,  earned 
by  caring  for  furnace,  20  $ ;    Nov.  6,  earned  by  cleaning 
automobile,  25$ ;  Nov.  7,  earned  by  caring  for  furnace,  15$. 
Payments:  Nov.  7,  cap,  75$.    Balanfted  Nov.  8. 

4.  Receipts:  May  1,  cash  on  hand,  $38.75 ;  May  3,  J.  C- 
Williams,  $20;  May  5,  R.  S.  James,  $36.85.    Payments: 
May  4,  groceries,  $8.75;  meat,  $2.80.    Balanced  May  6. 

5.  Receipts:  May  9,  cash  on  hand,   $275.25;  May  10, 
R.  J.  Benjamin,  $73.75;    May  12,  S.  K.  Henry,  $250.75. 
Payments:  May  9,  rent,  $85;  May  12,  clerks,  $75;  May  15, 
coal,  $22.50  ;  May  17,  account  book,  $1.50 ;  May  22,  tele- 
phone, $2.75;  May  23,  gas,  $5.60.    Balanced  May  23. 

6.  Receipts:    Aug.  1,  cash  on  hand,  $178.50;  Aug.  2, 
P.  M.  Myers,  $87.60;  Aug.  3,  A.  B.  Noyes,  $49.75;  Aug.  5, 
R.  L.  Dow,  $78.80.    Payments :  Aug.  2,  rent,  $50  ;  Aug.  3, 
electric  light,  $9.75  ;  Aug.  4,  gas,  $4.80  ;  Aug.  5,  J.  P.  Sin- 
clair, wages,  $16.75 ;   Aug.  5,  telephone,  $2.50 ;   Aug.  6, 
M.  R.  Roe,  wages,  $15.    Balanced  Aug.  7. 

Problems  of  this  kind  should  be  made  up  by  the  students,  who 
should  be  urged  to  keep  their  personal  accounts. 


ARITHMETIC  OF  THE  HOME 


Household  Account.    The  following  will  be  found  a  con- 
venient form  for  keeping  an  account  of  household  expenses : 


1^20 


RECEIPTS 


PAYMENTS 


skflt. 


8 


10 


16 


/6 


2  da. 


73 


85 


qg 


O/ 


/8 
2 


2 


85 


20 

y-o 

26 
35 
35 
30 

y-o 

50 
75 
50 


Of 


The  balance  should  agree  with  the  cash  on  hand.  The 
balance  is  found  by  adding  the  items  of  payments  and  sub- 
tracting this  sum  from  the  sum  of  the  receipts.  In  this 
case  we  have  |85.92  —  $39.91.  The  sum  of  the  receipts 
will  then  agree  with  the  sum  of  the  payments  and  the 
balance.  Bookkeepers  usually  write  the  balance  in  red. 


HOUSEHOLD  ACCOUNTS  5 

Exercise  2.    Household  Accounts 

1.  Extend  the   account   on   page  4  from   Sept.  16    to 
Sept.  24  by  including  the  following  receipts  and  payments : 
Receipts :  Sept.  18,  Mrs.  Adams,  loan  repaid,  $3.50  ;  house- 
hold allowance,  $20.  Payments:  Sept.  18,  telephone,  $1.75; 
vegetables,  650;    Sept.  19,  trolley  tickets,  $1;   shoes,  $4; 
ribbon,  750;  Sept.  21,  umbrella,  $2.50;  belt,  750;  Sept.  23, 
laundry,  900;  charity,  $1.50;  groceries,  $5.15;  ice,  450. 

Make  out  and  balance  the  following  accounts : 

2.  Receipts:    Oct.  9,  cash  on  hand,   $15.42;    Oct.  10, 
allowance,  $15 ;  Oct.  12,  loan  repaid  by  Mr.  Green,  $5.80. 
Payments:  Oct.  10,  insurance,  $2.50  ;  help,  $1.50;  lecture, 
500;  Oct.  11,  dress  goods,  $5.80;  trimming,  $1.10;  Oct.  12, 
raincoat,  $6.50 ;  shoes  repaired,  $1.25 ;  Oct.  13,  groceries, 
$3.90;   milk,  $2.80;  Oct.  15, 'carfare,  300;   flowers,  400; 
dressmaker,  3  da.  @  $1.80.    Balanced  Oct.  16. 

3.  Receipts:  Nov.  18,  cash  on  hand,  $12.67;  Nov.  19, 
allowance,  $22.50.    Payments:   Nov.  19,  cleaning  gloves, 
250 ;  dress  goods,  $9.60  ;  Nov.  20,  telephone,  $1.90  ;  remov- 
ing garbage,  750;  Nov.  21,  gas,  $2.30;  electric  light,  $1.80; 
magazines,  300;  Nov.  22,  books,  $2.60;  groceries,  $7.30; 
meat,  $1.20 ;  charity,  $1.25.    Balanced  Nov.  23. 

4.  Write  an  account  setting  forth  the  reasonable  expenses 
of  a  week  for  a  family  of  two  adults  and  three  children,  to 
include  groceries,  meat,  milk,  gas,  electric  light,  telephone, 
and  such  other  items  as  would  be  found  in  an  account  of 
such  a  family  in  your  vicinity.   Take  the  balance  on  hand 
as  $4.80  and  the  weekly  allowance  for  household  expenses, 
exclusive  of  rent  and  clothes,  as 


For  Ex.  4  the  students  should  be  asked  to  make  inquiry  at  home 
as  to  prices  and  reasonable  purchases. 


6  ARITHMETIC  OF  THE  HOME 

Need  for  knowing  about  Per  Cents.  John  Adams  is 
manager  of  the  baseball  team  of  a  junior  high  school. 
He  finds  that  he  can  buy  4  bats  at  500  each.  The  next 
day  he  sees  that  they  have  been  marked  down  10  per  cent. 
Hence  he  should  know  what  per  cent  means. 

It  is  probable  that  .everyone  in  the  class  knows.  In  that  case  some 
student  may  tell  the  meaning;  otherwise  the  subject  may  be  dis- 
cussed after  page  7  has  been  read.  This  page  is  merely  a  preparatory 
reading  lesson  for  those  who  may  not  already  have  studied  percent- 
age, which  subject  will  now  be  considered. 

John's  mother  wishes  to  buy  a  dress  for  his  sister.  The 
price  was  $12,  but  the  dress  has  been  marked  down  15  per 
cent.  If  she  wishes  to  buy  the  dress,  she  should  know 
what  this  means.  Do  you  know  what  it  means  ? 

The  teacher  has  to  make  a  report  of  the  number  of 
students  tardy  or  absent  last  week.  She  says  in  the  report 
that  4  per  cent  of  the  students  were  tardy  and  3  per  cent 
were  absent.  Do  you  know  what  this  means? 

There  were  10  questions  on  an  examination  in  arithmetic 
and  a  boy  answered  90  per  cent  of  them  correctly.  Do  you 
know  how  many  questions  he  answered  correctly  ?  Do 
you  know  how  many  he  did  not  answer  correctly? 

A  man  wishes  to  buy  an  automobile.  He  can  buy  a 
new  one  of  the  kind  he  likes  for  $1200,  but  one  of  the 
same  make  that  is  nearly  new  is  offered  for  30  per  cent 
less  than  this  price.  Do  you  know  how  to  find  what  the 
man  would  have  to  pay  for  the  second-hand  car? 

Do  you  know  the  meaning  of  the  symbols  10%,  15%, 
4%,  3%,  90%,  and  30%  ? 

It  is  not  necessary  that  any  student  should  be  able  to  answer  these 
questions.  What  is  of  importance  is  that  each  member  of  the  class 
should  see  that  per  cents  are  frequently  encountered  and  that  every- 
one must  know  what  they  mean. 


PER  CENTS  7 

'  Per  Cent.  Another  name  for  "  hundredths  "  is  per  cent. 
For  example,  instead  of  saying  "  ten  hundredths  "  we  may 
say  "  ten  per  cent."  The  two  expressions  mean  the  same. 
If  John  Adams  finds  that  some  bats  which  are  marked 
$2  can  be  bought  for  10  per  cent  less  than  the  marked 
price,  this  means  that  they  can  be  bought  for  JJ/Q,  or  -j^, 
less  than  $2  ;  that  is,  they  can  be  bought  for  f  2  less  ^  of 
$2,  or  |2-$0.20,  or  $1.80. 

Symbol  for  Per  Cent.  There  is  a  special  symbol  for  per 
cent,  %.  Thus  we  write  20%  for  20  per  cent,  or  0.20. 

Hundredths  written  as  Per  Cents.  Because  hundredths 
and  per  cents  are  the  same,  any  common  fraction  with  100 
for  its  denominator  may  easily  be  written  as  per  cent.  Thus 


Exercise  3.   Reading  Per  Cents 

All  work  oral 
Read  the  following  as  per  cents  : 

4'  *•         7'  *•         1°-  *. 


3.  rfo-   6-  *«   9-  *•    12-  *•    i5 

Read  the  following  as  hundredths  : 

16.  6%.      18.  22%.      20.  43%.     22.  66%.     24.  100%. 

17.  9%.      19.  37%.      21.  50%.     23.  72%.     25.  300%. 
26.  Read  |^  as  per  cent,  and  125%  as  hundredths. 


6J    16|     i     874      f   87i 
27.  Read  as  per  cents:  -JL,  ^  ^,  _g,  JL, 


8  ARITHMETIC  OF  THE  HOME 

Important  Per  Cents.  Mr.  Fuller  says  that  he  sold  a 
used  car  for  50%  of  what  it  cost  him.  He  sold  the  car 
for  how  many  hundredths  of  what  it  cost  him?  Express 
the  answer  as  a  common  fraction  in  lowest  terms. 

If  a  baseball  team  played  36  games  and  lost  25%  of 
them,  how  many  hundredths  of  the  games  did  it  lose  ? 
Express  the  answer  as  a  common  fraction  in  lowest  terms. 

From  these  two  examples,  and  by  dividing  25%  by  2, 
we  see  that 

50%  =4  25%  =  J  124%  =4 

If  this  circle  is  called  100,  how  much  is  the  shaded 
part?  What  per  cent  of  the  circle  is  the  shaded  part? 

Read  0.33^,  using  the  words  "  per  cent  "  ; 
using  the  word  "  hundredths."  How  many  thirds 
are  there  in  1  ?  How  many  times  is  33^%  con- 
tained in  1?  Then  33^%  is  what  part  of  1  ? 

From  these  two  examples,  and  by  taking  2  x  0.33^  and 
^-  of  0.33^,  we  see  that 

33J%=i  66f%=|  16f%=4 

Read  0.2  and  0.20,  using  the  word  "  hundredths  "  in 
each  case  ;  using  the  words  "  per  cent."  What  relation 
do  ybu  see  between  0.2  and  0.20?  between  0.2  and  20%? 
between  fa  and  20%? 

What  is  the  relation  of  ^  tp  ^?   to  40%  ? 

What  is  the  relation  of  0.60  to  J-  ?   of  0.80  to  f  ?  • 

From  answers  to  these  questions  we  see  that 
20%  =1  40%=  |  60%=  |  80%  =4 

In  the  same  way  it  is  easily  seen  that. 
75%=  |  37£%=f  62J%=| 


Such  per  cent  equivalents  should  be  drilled  upon  so  thoroughly 
that  the  mention  of  one  form  automatically  suggests  the  other. 


IMPORTANT  PER  CENTS  9 

Exercise  4.    Finding  Per  Cents 

All  work  oral 

1.  A  woman  who   had   set  aside   $16   for  household 
expenses  for  a  week  finds  that  she  has  spent  50%  of  it. 
How  much  money  has  she  spent? 

Find  50%  of  each  of  the  following  numbers: 

2.  18.  3.  20.  4.  36.  5.  64.  6.  400. 

7.  If  this  square  represents  a  box  cover  which  contains 
100  sq.  in.,  how  many  square  inches  are  shaded  ? 

How  many  fourths  of  the  cover  are  shaded  ?  How 
many  hundredths  of  the  cover  are  shaded? 
What  per  cent  of  the  cover  is  shaded? 

8.  A  man  with  an  income  of  $4  a  day  spends  25%  of 
it  for  food.     How  much  money  does  he  spend  for  food  ? 

Find  25°/0  of  each  of  the  following  numbers : 

9.  16.          10.  36.          11.  48.          12.  5.          13.  240. 

Find  the  values  of  the  following : 

14.  20%  of  75.      17.  80%  of  45.  20.  75%  of  800. 

15.  40%  of  25.      18.  33£%  of  75.  21.  87£%  of  1600. 

16.  60%  of  35.      19.  16f  %  of  66.  22.  66f  %  of  6000. 

23.  If  you  answer  correctly  66|-%  of  the  questions  on 
an  examination  paper  and  there  are  12  questions  in  all, 
how  many  questions  do  you  answer  correctly  ? 

24.  If  a  boy  is  at  bat  10  times  and  makes  base  hits 
20  %  of  the  times,  how  many  base  hits  does  he  make  ? 

25.  If  a  merchant  gains  33^%  on  $1500,  what  fractional 
part  of  $1500  does  he  gain  ?    How  much  does  he  gain  ? 


10  ARITHMETIC  OF  THE  HOME 

Per  Cents  and  Common  Fractions.  We  have  learned  that 


~100~200~8 

To  express  per  cent  as  a  common  fraction,  write  the  number 
indicating  the  per  cent  for  the  numerator  and  100  for  the 
denominator,  and  then  reduce  to  lowest  terms. 

It  is  sometimes  convenient  to  use  one  form,  and  some- 
times another.  Thus,  if  we  are  multiplying  by  26.9%,  it  is 
easier  to  think  of  the  multiplier  as  0.269  ;  but  if  we  are 
multiplying  by  66|-%,  it  is  easier  to  think  of  it  as  ^. 

We  know  that  ^  may  be  reduced  to  hundredths  by 
multiplying  each  term  by  33^.  We  then  have 


To  express  a  common  fraction  as  per  cent,  reduce  it  to 
a  common  fraction  with  100  for  the  denominator,  and  then 
write  the  numerator  followed  by  the  symbol  for  per  cent. 

We  may  also  reduce  ^  to  hundredths  by  dividing  2  by  3,  thus : 
1  =  2  +  3  =  2.00  H-  3  =  0.66§  =  66f  %. 

Exercise  5.   Per  Cents  and  Common  Fractions 

Express  as  common  fractions  in  lowest  terms: 

1.  12%.     3.  36%.     5.  64%.      7.  29%.       9..1?1%. 

2.  24%.     4.  3J%.     6.  32%.     8.  45%.     10.  66.6|%. 

Express  the  following  as  per  cents : 

11.  f.        13.  f.        15.  f        17.  f.        19.   &.        21.    <£}. 

12.  f.        14.  f.        16.  f.        18.  f        20.  ^.        22.  ^ 


PEK  CENTS  AS  DECIMALS  11 

Per  Cents  as  Decimals.  Since  25.5%  and  0.255  have  the 
same  value,  we  see  the  truth  of  the  following: 

To  express  as  a  decimal  a  number  written  with  the  per  cent 
sign,  omit  the  sign  and  move  the  decimal  point  two  places  to 
the  left,  prefixing  zeros  if  necessary. 

When  we  omit  the  per  cent  sign  we  must  indicate  the  hundredths 
in  some  other  way,  as  by  moving  the  point  two  places  to  the  left. 
Thus  2£%  =  0.021,  or  0.0225  ;  325%  =  3.25  ;  0.7%  =  0.007. 

Exercise  6.   Per  Cents  as  Decimals 

Examples  1  to  13,  oral 

1.  If  3%  of  the  students  of  this  school  are  absent  to-day 
how  many  are  absent  out  of  every  100  ?  out  of  every  200  ? 

2.  How  much  is  ^  of  $400?    1%  of  |400  ? 

3.  How  much  is  0.05  of  $100?   5%  of  $100?    5%  of 
$200?   5%  of  $2000?    5%  of  $4000? 

Read  as  decimals,  whole  numbers,  or  mixed  decimals : 

4.  35%.     6.  27%.     8.  325%.     10.  365%.     12.  400%. 

5.  72%.     7.  42%.     9.  225%.     11.  425%.     13.  200%. 

14.  Express  ^%  as  a  decimal;  as  a  common  fraction. 

15.  Express  -|%  as  a  decimal;  as  a  common  fraction. 

16.  Express  666|-%  as  a  decimal ;  as  an  improper  fraction. 

17.  How  much  is  0.35  of  $300?    35%  of  $300? 

18.  How  much  is  2|  x  $650  ?    2.75  x  $650  ?    275%  of 
$650?    3Jx$750?    3.25  x  $750?    325%  of  $750? 

Express  as  decimals,  whole  numbers,  or  mixed  decimals: 

19.  24%.    21.  33£%.    23.  300%.    25.  3000%. 

20.  36%.    22.  0.8%.    24.  250%.    26.  0.008%. 


12  AKITHMETIC  OF  THE  HOME 

Decimals  as  Per  Cents.  Since  per  cent  means  hundredths, 
to  express  a  decimal  as  per  cent  we  have  to  consider  only 
how  many  hundredths  the  decimal  represents. 

1.  Express  0.5  as  per  cent. 

Since  0.5  =  0.50,  we  see  that  0.5  is  the  same  as  50  hundredths, 
or  50%. 

2.  Express  0.625  as  per  cent. 

Since  0.625  =  62.5  hundredths,  or  62^  hundredths,  we  see  that 
0.625  is  the  same  as  62.5%,  or  62£%. 

3.  Express  0.00375  as  per  cent. 

Since  0.00375  =  0.00Ty^  =  O.OOf ,  we  see  that  0.00375  =  f  %. 

4.  Express  4.2-|  as  per  cent. 

Since  4.2£  =  f  §£,  we  see  that  4.2£  =  425%. 

Therefore,  to  express  a  decimal  as  per  cent,  write  the  per 
cent  sign  after  the  number  of  hundredths. 

Exercise  7.   Decimals  as  Per  Cents 

Examples  1  to  6,  oral 

1.  A  foot,  being  0.33^  yd.,  is  what  per  cent  of  a  yard  ? 

2.  A  peck,  being  0.25  bu.,  is  what  per  cent  of  a  bushel  ? 

3.  A  quart,  being  0.125  pk.,  is  what  per  cent  of  a  peck? 

4.  Express  0.24  mi.  as  per  cent  of  a  mile. 

5.  Express  1  oz.  as  per  cent  of  10  oz. ;  of  10Q  oz. 

6.  Express  0.2ft.  as  per  cent  of  1ft.;  of  2ft. 

Express  the  following  as  per  cents : 

7.  0.7.  9.  0.42.  11.  0.625.  13.  6.66f. 

8.  0.8.  10.  0.39.  12.  0.375.  14.  0.00875. 


FINDING  PER  CENTS  13 

Finding  Per  Cents.  We  have  now  found  that  certain 
important  fractions  like  -|,  ^,  ^,  -J,  -|,  and  -£•  are  easily 
expressed  as  per  cents ;  that  any  fraction  can  be  expressed 
as  a  per  cent  by  first  reducing  it  to  hundredths ;  and  that 
per  cents  can  easily  be  expressed  as  common  fractions. 

If  we  wish  to  find  50%,  25%,  12^%,  33J%,  16f%, 
or  20%  of  a  number,  it  is  easier  to  use  the  equivalent 
common  fraction,  as  in  the  first  of  these  problems: 

1.  If  Kate  wishes  to  buy  a  suit  that  is  marked  $12, 
and  finds  that  it  is  to  be  marked  down  12-|%,  how  much 
will  the  suit  cost  her  after  it  is  marked  down  ? 

Since  12^%  =  ^,  we  need  simply  to  take  ^  of  $12,  and  this  is  the 
amount  the  suit  is  to  be  marked  down. 

Then  we  have  1  of  $12  =  $1.50 ; 

and  $12  -  $1.50  =  $10.50,  cost. 

2.  Robert  wishes  a  Boy  Scout  suit.    The  marked  price 
of  the  suit  he  wants  is  $7.50,  but  he  finds  that  to-morrow 
at  a  bargain  sale  this  suit  is  to  be  marked  down  15%.   How 
much  will  it  then  cost  him? 

Since  15%  =  0.15  we  should  find 
0.15  of  $7.50.  We  then  have  17.50  $7.50 

0.15  of  $7.50  =  $1.125, 
and    $7.50  -  $1.13  =  $6.37,  cost. 

The  dealer  will  probably  de- 
duct only  $1.12,  neglecting  the  5, 
in  order  to  make  the  computation 
easier  and  to  take  advantage  of 

the  half  cent  himself.  There  is  no  general  custom  as  to  using  a  frac- 
tion of  a  cent.  The  student  should  consider  ^  or  more  as  a  whole 
cent  in  each  operation  except  in  cases  of  discount. 

To  find  a  given  per  cent  of  a  number,  express  the  per  cent 
as  a  common  fraction  or  as  a  decimal  and  multiply  the  num- 
ber by  this  result. 


3750       $6.38,  cost 


$1.1250 


14  ARITHMETIC  OF  THE  HOME 

Exercise  8.   Finding  Per  Cents 
Examples  1  to  20,  oral 

1.  A  man  spends  20%  of  his  income  for  rent.    What 
fraction  of  his  income  does  he  spend  for  rent? 

2.  A  man  saves  10%   of  his  income.     His  income  is 
$90  a  month.    How  much  does  he  save  in  a  month? 

3.  A  girl  spends  for  clothes  25%    of  her  allowance. 
Express  this  per  cent  as  a  common  fraction. 

Express  as  common  fractions  or  as  whole  or  mixed  numbers : 

4.  7%.        7.  75%.      10.  1%.      13.  100%.      16.  125%. 

5.  30%.     8.  60%.      11.  2%.      14.  200%.     17.  250%. 

6.  40%.     9.  90%.      12.  4%.      15.  500%.      18.  375%. 

19.  How  much  is  10%  of  420  lb.?  of  $250  ?  of  30  yd.? 

20.  Milk  yields  in  butter  about  4%  of  its  weight.    How 
much  butter  will  25  lb.  of  milk  yield  ? 

21.  If  a  merchant  pays  1^0  apiece  for  pencils  and  sells 
them  at  a  profit  of  50%   on  the  cost  price,  what  is  the 
selling  price  per  hundred  ? 

22.  If  a  butcher  buys  a  certain  kind  of  meat  at  15$  a 
pound  and  sells  it  at  a  profit  of  20%  on  the  cost  price,  at 
what  price  does  he  sell  it  per  pound  ? 

23.  If  a  dealer  pays  7^<k  a  quart  for  milk  and  sells  it  at 
a  profit  of  20  %  on  the  cost  price,  how  much  does  he  receive 
for  200  qt.  of  milk  ? 

Find  the  values  of  the  following  : 

24.  30%.  of  2400.  27.  75%  of  $6280. 

25.  16%  of  $3600.  28.  27%  of  3700ft. 

26.  35%  of  $825.  29.  3£%  of  $4275. 


HOME  PROBLEMS  15 

Exercise  9.   Reading  the  Gas  Meter 

1.  Here  is  a  picture  of  the  three  dials  on  a  gas  meter. 
The  left-hand  dial  indicates  ten  thousands,  the  middle 
dial  indicates  thousands,  and  the  right-hand  dial  indicates 


hundreds.  The  dial  shows  that  64,300  cu. 'ft.  of  gas  has 
passed  through  the  meter.  If  this  is  the  reading  for  May  1, 
and  the  reading  for  April  1  was  62,300,  how  much  was 
the  gas  bill  for  April  at  80  <£  per  M  (1000  cu.  ft.)? 

In  the  above  case  more  than  64,300  cu.  ft.  has  passed  through  the 
meter,  but  we  read  only  to  the  hundred  last  passed.  Notice  that  the 
middle  dial  is  read  in  the  opposite  direction  from  the  others. 

2.  Read  this  meter.  If  the  reading  was  52,700  a  month 
ago,  how  much  is  the  gas  bill  for  the  month  at  $1  per  M  ? 


3.  If  the  gas   consumed  by  a  family  in  January  was 
3200  cu.  ft.,  and  that  consumed  in  June  was  10%   less 
than  in  January,  how  much  was  the  gas  bill  for  each  of 
these  months  at  $1.25  per  M  ? 

4.  A  certain  gas  company  deducts  10%  from  bills  paid 
before  the  tenth  of  the  month.    If  the  readings  are  72,400 
and  74,200  on  the  first  days  of  two  consecutive  months, 
and  the  rate  for  gas  is  $1.30  per  M,  how  much  will  be 
saved  by  paying  the  bill  before  the  tenth  of  the  month  ? 


16  AKITHMETIC  OF  THE  HOME 

Exercise  10.   Reading  the  Electric  Meter 

1.  Mr.  Jacobs  lights  his  store  by  electricity.  The  elec- 
tricity is  measured  in  kilowatt  hours  (K.W.H.),  and  the 
meter  shows  thousands,  hundreds,  tens,  and  units.  He 
reads  the  meter  in  a  way  similar  to  that  in  which  he 
reads  the  gas  meter.  Notice  that  the  second  and  fourth 
dials  are  read  clockwise  and  that  the  first  and  third  are 
read  counterclockwise.  Read  the  meter  shown: 


The  technical  meaning  of  the  K.W.H.  in  the  science  of  electricity 
need  not  be  considered  in  the  mathematics  of  the  junior  high  school. 

2.  If  Mr.  Jacobs  in  June  uses  40  K.W.H.  at  15$  and 
5  K.W.H.  at  80,  how  much  is  his  bill  for  the  month? 

Companies  charge  large  users  differently  according  to  the  number 
of  lights  or  machines.  For  example,  in  most  places  10  lights  would 
cost  more  in  proportion  than  200  lights. 

3.  If  Mr.  Jacobs  in  the  month  of  March  uses  in  his  house 
18  K.W.H.  at  140  and  has  a  reduction  of  4%  ( on  the  bill  if 
paid  before  the  15th  of  April,  how  much  is  his  bill  if  he 
takes  advantage  of  the  reduction  ? 

Find  the  amount  of  each  of  the  following  bills : 

4.  12  K.W.H.   at  150,   3  K.W.H.   at  70,   less  £0  per 
K.W.H.  on  account  of  prompt  payment. 

5.  28  K.W.H.  at  160,  6  K.W.H.  at  100,  less  £0  per 
K.W.H. 

The  student  should  actually  read  the  electric  meter  in  the  school 
or  at  his  home  if  this  can  be  done  conveniently. 


PEOBLEMS  OF  PEECENTAGE  17 

Three  Important  Problems  of  Percentage.  There  are  three 
important  problems  of  percentage.  The  first,  which  is  by 
far  the  most  important  one,  is  that  of  finding  a  given  per 
cent  of  a  number.  This  has  already  been  considered.  The 
second  problem  of  importance  is  to  find  what  per  cent  one 
number  is  of  another,  and  this  will  be  considered  on  page  18. 
The  third  problem  of  importance  is  to  find  the  number  of 
which  a  given  number  is  a  given  per  cent,  and  this  will 
be  considered  on  page  21. 

The  problems  on  pages  18  and  21  depend  upon  the 
following  principle: 

Griven  the  product  of  two  factors  and  one  of  the  factors,  the 
other  factor  may  be  found  by  dividing  the  product  by  the 
given  factor. 

That  is,  if  we  have  given  10,  the  product  of  2  and  5,  we 
can  find  the  factor  2  by  dividing  the  given  product  by  5 
and  the  factor  5  by  dividing  the  product  by  2. 

In  this  review  of  percentage  we  shall  confine  the  applied  problems 
largely  to  those  relating  to  home  interests,  but  shall  occasionally 
introduce  other  problems  for  the  purpose  of  variety. 

Exercise  11.    Product  and  One  Factor 

All  work  oral 

The  first  number  in  each  of  the  following  examples  being 
the  product  of  two  factors,  and  the  second  number  being  one 
of  these  factors,  find  the  other  factor : 

1.  72,  8.         4.  36,  9.  7.  90,  9.  10.  300,  10. 

2.  72,  9.          5.  80,  10.         8.  60,  6.  11.  9,  3. 

3.  36,  4.         6.  80,  8.  9.  300,  30.        12.  T9¥,  3. 

13.  I  am  thinking  of  the  number  which,  multiplied  by  9, 
gives  the  product  63.  What  is  the  number? 


18  ARITHMETIC  OF  THE  HOME 

Finding  what  Per  Cent  One  Number  is  of  Another.  The 
second  problem  of  importance  in  percentage  mentioned  on 
page  17  is  to  find  what  per  cent  one  number  is  of  another. 

For  example,  if  in  a  certain  test  Anna  solved  correctly 
8  problems  out  of  12,  what  per  cent  of  the  problems  did 
she  solve  correctly? 

Here  we  have  a  certain  per  cent  x  12  =  8 ;  that  is,  we  have  the 
product  (8)  and  one  factor  (12),  to  find  the  other  factor  (a  certain 
per  cent).  Therefore 

8  •*•  12  =  0.66f  =  665%,  the  per  cent  solved  correctly. 
\ 

Exercise  12.  'Product  and  One  Factor 

All  work  oral 

1.  If  there  are  30  students  enrolled  in  a  class  and  3  of 
them  are  absent  to-day,  what  per  cent  are  absent  ? 

2.  On  an  automobile  trip  of  60  mi.,  what  per  cent  of  the 
distance  has  a  man  made  when  he  has  traveled  30  mi.  ? 

3.  During  a  series  of  games  Fred  was  at  bat  36  times 
and  made  9  base  hits.    What  was  his  batting  average  ? 

4.  If  a  baseball  team  wins  14  games  out  of  20  games 
played,  what  per  cent  of  the  games  does  it  win  ? 

Find  what  per  cent  the  second  number  is  of  the  first : 

5.  4,  2.  9.  6,  2.  13.  45,  15.         17.  64,  8. 

6.  4,  3.  10.  40,  30.         14.  75,  15.         18.  64,  16. 

7.  4,  4..          11.  30,  15.         15.  68,  34.         19.  36,  18. 

8.  5,  2J.        12.  90,  45.         16.  72,  36.         20.  15,  7J. 

21.  If  a  book  has  240  pages,  what  per  cent  of  the  num- 
ber of  pages  have  you  read  when  you  have  read  through 
page  24  ?  when  you  have  read  through  page  48  ?  when 
you  have  read  through  page  72  ?  through  page  1 20  ? 


PEOBLEMS  OF  PERCENTAGE  19 

Application  to  Written  Exercises.  You  now  understand 
the  second  important  problem  in  percentage.  We  shall 
consider  once  more  the  first  problem,  and  then  the  appli- 
cation of  the  second  to  written  exercises. 

1.  A  man  bought  an  automobile  for  $800  and  sold  it 
at  a  profit  of  25%  on  the  cost.    How  much  did  he  gain? 

Here  we  have  to  find  25%  of  $800,  or  0.25  of  $800 ;  that  is,  we 
have  two  factors  given,  0.25  and  $800,  to  find  the  product.  Therefore 

we  have  0.25  x  $800  =  $200. 

That  is,  the  man  gained  $200. 

2.  A  man  bought  an  automobile  for  $800  and  sold  it  at 
a  profit  of  $200.    What  per  cent  of  the  cost  did  he  gain  ? 

Here  we  have  $200  equal  to  some  per  cent  of  $800 ;  that  is,  we 
have  given  the  product  ($200)  of  two  factors  and  one  of  the  factors 
($800),  to  find  the  other  factor.  Therefore  we  have 

$200  •*-  $800  =  0.25  =  25%. 
That  is,  the  man  gained  25%  of  the  cost. 

Exercise  13.   Product  and  One  Factor 

1.  $260  is  what  per  cent  of  $5200?   of  $7800? 

2.  $22.50  is  what  per  cent  of  $450?   of  $67.50? 

3.  $20.24  is  what  per  cent  of  $506  ?   of  $404.80  ? 

4.  $58.20  is  what  per  cent  of  $931.'20  ?   of  $349.20? 

5.  A  foot  is  what  per  cent  of  a  yard  ?  of  2  yd.  ?  of  8  yd.  ? 

6.  A  quart  is  what  per  cent  of  a  gallon  ?    of  16  gal.  ? 

7.  4  is  what  per  cent  of  •§•  ?   If-  is  what  per  cent  of  £  ? 

rr  •*•  O  O  •*•  T= 

8.  35%  is  what  per  cent  of  70%?  of  140%?  of  210%? 

9.  33J%  is  what  per  cent  of  66f  %  ?  of  1  ?  of  133J%  ? 
10.  66f%  is  what  per  cent  of  33J%?  of  66f%?  of  2  ? 


20  ARITHMETIC  OF  THE  HOME 

11.  If  an  employer  reduces  the  working  day  for  his  men 
from  9  hr.  to  8  hr.,  what  is  the  per  cent  of  reduction  ? 

12.  If  a  class  had  20  examples  to  solve  on  Monday  and 
28  on  Tuesday,  what  was  the  per  cent  of  increase  ? 

13.  If  you  have  read  78  pages  in  a  book  of  300  pages, 
what  per  cent  of  the  pages  have  you  read? 

14.  A   man's  income  is   $3800   a  year  and  he   spends 
$1914.    What  per  cent  of  his  income  does  he  save  ? 

15.  In  a  certain  city  1152  out  of  the  2400  pupils   in 
school  are  boys.    What  per  cent  are  boys  ? 

16.  In  a  certain  school  360  out  of  750  pupils  are  girls. 
What  per  cent  are  girls  ? 

17.  If  a  class  devotes  42  min.  a  day  to  arithmetic  one 
year,  and  45  min.  a  day  the  next  year,  what  is  the  per  cent 
of  increase  per  day  ? 

18.  If  the  last  chapter  of  a  book  is  numbered  XXXV 
and  you  have  finished  reading  Chapter  VII,  what  per  cent 
of  the  chapters  have  you  read  ? 

19.  The  purity  of  gold  is  measured  in  carats,  or  twenty- 
fourths,  18  carats,  or  18  carats  fine,  meaning  ^-|  pure  gold. 
What  is  the  per  cent  of  pure  gold  in  a  14-carat  ring  ? 

20.  What  is  the  per  cent  of  pure  gold  in  a  watch  case 
that  is  18  carats  fine  ?  in  a  chain  that  is  10  carats  fine  ? 
in  an  ingot  of  pure  gold  ? 

21.  If  3  members  of  a  class  of  48  have  not  been  either 
tardy -or  absent  during  the  year,  and  6  members  have  not 
been  absent,  what  per  cent  have  not  been  either  tardy  or 
absent  ?  What  per  cent  have  not  been  absent  ? 

22.  Stockings  are  marked  35$  a  pair  or  3  pairs  for  $1. 
What  per  cent  on  the  higher  price  does  a  customer  save 
who  buys  3  pairs  for  $1  instead  of  paying  35$  a  pair? 


PROBLEMS  OF  PERCENTAGE  21 

Finding  the  Number  of  which  a  Given  Number  is  a  Given 
Per  Cent.  The  third  important  problem  in  percentage 
mentioned  on  page  17  is  to  find  the  number  of  which  a 
given  number  is  a  given  per  cent. 

For  example,  if  a  baseball  team  has  lost  9  games,  which 
is  45%  of  the  number  of  games  played,  how  many  games 
has  the  team  played? 

Here  we  have  given  the  fact  that  45%  of  the  number  of  games 
is  9.  In  other  words,  we  have  given  the  product  (9)  of  two  factors 
and  one  of  the  factors  (0.45),  to  find  the  other  factor. 

Therefore  we  have  9  -r-  0.45  =  20. 

That  is,  the  team  has  played  20  games. 
Check.  45%  of  20  =  9. 

Exercise  14.    Product  and  One  Factor 

Examples  1  to  13,  oral 

1.  If  20%  of  the  inhabitants  of  a  certain  city  are  school 
children  and  there  are  20,000  school  children  in  the  city, 
what  is  the  total  population  of  the  city  ? 

2.  If  23%  of  the  inhabitants  of  a  certain  city  voted  at  a 
certain  election  and  there  were  2300  ballots  cast,  what  was 
the  total  population  of  the  city  ? 

3.  A  player  reached  first  base  20  times,  or  33^%  of  the 
times  he  was  at  bat.    How  many  times  was  he  at  bat  ? 

4.  An  automobile  was  sold  second-hand  for  $480,  which 
was  40%  of  the  amount  paid  for  it  originally.    How  much 
was  paid  for  it  originally  ? 

5.  If  you  pay  30%  of  tiie  expenses  of  a  camping  trip 
and  pay  $12,  what  are  the  expenses  of  the  trip  ? 

6.  In  a  certain  school  there  are  170  boys,  which  is  85% 
of  the  number  of  girls.    How  many  girls  are  there  ? 


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HOME  ACCOUNTS 


23 


Exercise  15.   Expense  Accounts 

In  ike  foUoidng  family  expense  account  for  4  mo-,  the 
interne  being  $125  a  montk,  foul: 

1.  The  amount  saved  each  month. 

2.  The  total  of  each  hem  for  4  mo.,  inclnding  savings. 

3.  The  per  cent  which  each  total  is  of  the  grand  total 


EXPENSE  Accoryr 

FEB. 

H  .  i 

Ar*. 

TOTAL 

Food    

24.10 

23.30 

24.80 

22.15 

Household 

Rent     

18.00 

laoo 

18.00 

18.00 

Fuel  (average)  .     . 
Gas  (cooking)    .     . 
Electricity      .     .     . 
Help     

4.20 
2.20 
1.60 
1.80 

4.20 

4,20 
2.35 
L8fl 

2.25 
.65 

4.20 

2.15 
1.30 
1.30 
2.30 

4.20 
2.25 
1.45 
1.00 
9^60 

Furnishings  ... 
Personal 

Clothing    .... 
Carfare    .     .     .     . 

7.00 
.80 

2,00 

1.25 

.90 

24.00 
1.20 

Insurance 

Health  and  accident 

1.10 

L10 

1.10 

1.10 

Life  (average)    .     . 
Benevolences 

4.30 

4.30 

4.30 

4.30 

Church      .... 

3.70 

4.20 

2.60 

i£Q 

Charity      .... 
Education,  Recreation 

1.30 
1.90 

-.-:  I 
2.30 

.80 
4.70 

3.20 
3.60 

Incidentals    .... 

7.30 

1.30 

2.60 

7.38 

Savings    

Total     .... 

These  per  cents  AanW  be  carried  to  the  nearest  tenth. 


24  ARITHMETIC  OF  THE  HOME 

Problem  Data.  The  following  price  list  may  be  used  in 
solving  the  problems  on  page  25  and  similar  problems. 

The  data  may  also  be  secured  by  the  students  through  inquiry 
at  home  or  at  some  grocery.  This  list  may  be  made  the  basis  of  prac- 
tical problems  in  simple  domestic  bookkeeping.  The  object  is,  of 
course,  to  make  arithmetic  as  real  as  possible,  and  when  this  purpose 
has  been  served,  the  student  should  proceed  to  other  topics. 

Allspice,  10  $  per  can ;  $1  per  dozen  cans. 
Asparagus,  35  $  per  can ;  $4  per  dozen  cans. 
Bacon,  American,  280  per  pound. 

Sliced,  in  jars,  30$  per  pound;  $3.25  per  dozen  jars. 
Breakfast  cereal,  14  $  per  package ;  $1.60  per  dozen  packages. 
Cinnamon,  10  $  per  can ;  $1  per  dozen  cans. 
Cloves,  30  $  per  pound ;  50  $  per  2  pound  box. 
Cocoa,  half-pound  cans,  25  $;  $2.75  per  dozen  cans. 
Coffee,  Maracaibo,  20  $  per  pound ;  5  Ib.  for  85  $. 

Java  and  Mocha,  35  $  per  pound ;  5  Ib.  for  $1.60. 

Old  Government  Java,  green,  27$  per  pound ;  5  Ib.  for  $1.30. 
Crackers,  Salines,  25  $  per  tin ;  $2.75  per  dozen  tins. 

Ginger  snaps,  8  $  per  carton ;  90  $  per  dozen  cartons. 
Domino  sugar,  5  Ib.  for  60  $. 
Flour,  buckwheat,  6  $  per  pound ;  a  bag  of  24^  Ib.,  $1.30. 

Self-raising,  3  Ib.  for  19  $ ;  6  Ib.  for  35  $. 

Wheat,  5$  per  pound ;  $6  per  barrel  of  196  Ib. ;  90$  per  sack 

of  24|lb. 

Granulated  sugar,  8  $  per  pound. 
Herring,  15  $  per  can ;  $1.75  per  dozen  cans. 
Honey,  8-ounce  bottles,  30  $;  $3.25  per  dozen  bottles. 
Loaf  sugar,  11  $  per  pound. 

Macaroni,  12  $  per  package ;  25  packages  for  $2.75. 
Maple  sirup,  pints,  25$;  gallon  cans,  $1.45 ;  $16.50  per  dozen  cans. 
Olive  oil,  40  $  per  pint. 

Olives,  32  $  per  bottle ;  $3,.75  per  dozen  bottles. 
Soups,  half -pint  cans,  10  $;    pint  cans,  16  0;    quart  cans,  28$; 

$3.25  per  dozen  quart  cans. 

Sugar  sirup,  half-gallon  cans,  50$;  5-gallon  cans,  $4. 
Tea,  Black  India,  50$  per  pound. 

English  Breakfast,  48$  per  pound. 


HOUSEHOLD  ECONOMICS  25 

Exercise  16.    Household  Economics 

1.  If  a  family  wishes  a  dozen  cans  of  cocoa,  what  per 
cent  is  saved  in  buying  at  the  dozen  rate  ? 

In  such  cases  reckon  the  per  cent  on  the  higher  price. 

2.  If  a  family  wishes  5  gal.  of  sugar  sirup,  how  much 
is  saved  in  buying  a  5-gallon  can  instead  of  10  half -gallon 
cans  ?    What  per  cent  is  saved  ? 

3.  How  much  does  a  hotel  manager  save  in  buying 
120  gal.  of  maple  sirup  by  the  dozen  gallon  cans  instead 
of  by  the  single  can  ?    What  per  cent  does  he  save  ? 

4.  In  which  is  the  per  cent  of  saving  greater,  in  buying 
honey  by  the  dozen  bottles  instead  of  by  the  bottle,  or  in 
buying  maple  sirup  by  the  dozen  cans  instead  of  by  the  can? 

5.  If  a  woman  wishes  24^  Ib.  of  buckwheat  flour,  how 
much  does  she  save  in  buying  it  by  the  bag? 

6.  What  per  cent  is  saved  in  buying  self-raising  flour  by 
the  6-pound  package  instead  of  by  the  3-pound  package  ? 

7.  By  inquiry  at  home,  make  out  a  grocery  list  for  a 
week,  from  page  24.    Make  two  pages  of  a  home  account, 
the  left-hand  page  showing  the  amount  received,  and  the 
right-hand  page  showing  the  amounts  spent  for  groceries. 

8.  Make  out  a  bill  for  six  items  of  groceries,  making 
the  proper  extensions  and  footing.    Receipt  the  bill. 

Unless  the  students  recall  this  from  their  preceding  work  in  arith- 
metic, the  teacher  should  take  it  up  at  the  blackboard. 

9.  If  a  man  uses  2320  cu.  ft.  of  gas  in  April,  how  much 
is  his  gas  bill  for  that  month  at  80$  per  1000  cu.  ft.  ? 

10.  At  the  beginning  of  the  month  a  gas  meter  registers 
14,260,  and  at  the  end  of  the  month  17,140.  How  much 
is  the  gas  bill  for  the  month,  at  fl  per  1000  cu.  ft.? 


26  ARITHMETIC  OF  THE  HOME      "• 

Exercise  17.    Heating  the  House 

1.  A  man  put  a  hot-water  heater  in  his  house  at  a  cost 
of  $540,  and  found  that  he  used  12  T.  of  coal  last  season, 
the  coal  costing  $7.60  per  ton.    How  much  did  he  spend 
for  the  heater  and  fuel  ? 

2.  If  the  house  in  Ex.  1  was  heated  for  204  da.,  what 
was  the  average  cost  of  the  fuel  per  day  ? 

3.  If  the  house  in  Exs.  1  and  2  had  9   rooms,  what 
was  the  average  cost  of  the  fuel  per  room  per  day  ? 

4.  A  man  has  a  steam-heating  plant  in  his  house.    Last 
winter  it  consumed  22-£T.  of  coal  costing  $7.25  per  ton. 
How  much  did  the  coal  cost  ? 

5.  If  15%  of  the  coal  in  Ex.  4  was  lost  in  ashes,  how 
many  pounds  of  coal  were  lost  in  ashes? 

6.  If  the  house  in  Ex.  4  has  14  rooms  and  is  heated 
for  190  da.  in  a  year,  what  was  the  average  cost  of  the 
fuel  per  day  and  the  average  cost  per  room  per  day  ? 

7.  If  85%  of  the  weight  of  coal  is  used  in  producing 
heat  in  a  furnace,  how  many  tons  of  coal  are  transformed 
into  heat  by  a  furnace  that  burns  1 7  T.  in  a  season  ? 

8.  A  man  used  14  T.  of  coal  in  his  furnace  in  a  season, 
but  on  buying  a  new  furnace  he  used  8-£%  less  coal.    At 
$6.75  a  ton,  how  much  did  he  save  on  the  coal? 

9.  A  heating  plant  costing  $525  averages  12  T.  of  coal 
per  year  at  $7. 25 'a  ton  and  furnishes  the  same  amount  of 
heat  as  a  plant  costing  $375  and  averaging  14  T.  of  coal 
per  year  at  the  same  price.    Counting  as  part  of  the  cost 
an  annual  depreciation  of  10%  of  the  original  cost  price, 
and  not  considering  interest,  which  plant  costs  the  more 
money  in  4  yr.,  and  how  much  more  ? 


HOUSEHOLD  ECONOMICS  27 

Exercise  18.   The  Family  Budget 

1.  Last  year  Mr.  Stone  received  an  income  of  $3000. 
He  set  aside  certain  per  cents  of  his  income  as  follows: 
rent,   15%;  heat,    3%;    light,  1J%  ;  food,    28%;    wages, 
5-|%  ;    incidentals,    7%  ;    other   personal   expenses,  15%  ; 
books,    music,    church,    and    pleasure,    8%.     How   much 
money  did  Mr.  Stone  allow  for  each  of  these  purposes? 

2.  Mr.  Stone  in  Ex.  1  really  paid  for  rent,  $320;  for 
heat,  $52.75;  for  light,    $28.50;  for  food,  $608.75;  for 
wages,  $135;  for  incidentals,  $175.50;  for  other  personal 
expenses,    $302.80 ;   and    for   books,    music,    church,    and 
pleasure,  $167.75.    How  much  did  each  item  of  expendi- 
ture differ  from  the  estimate  and  how  much  did  Mr.  Stone 
save  during  the  year? 

3.  In  Ex.  2  what  per"  cent  of  the  amount  spent  for  rent 
and  food  was  spent  for  rent  and  what  per  cent  for  food  ? 

4.  Mr.  Sinclair  has   an  income   of  $175   a  month.    He 
pays  during  the  year  for  rent,  $480;  for  heat  and  light, 
$85.75;  for  food,  $675.80;  for  clothing,  $168.40;  for  in- 
surance, $54.75;  and  for  other  expenses,  $250.    What  per 
cent  of  his  income  does  he  save  ? 

5.  In  Ex.  4  what  per  cent  of  his  income  does  Mr.  Sinclair 
pay  for  rent  ?  for  food  ?    for  heat  and  light  ? 

6.  If  a  man's  income  is  $225  a  month  and  he  spends 
$600  a  year  for  rent,  what  per  cent  of  his  income  does  he 
spend  for  rent  and  what  per  cent  is  left  for  other  purposes  ? 

7.  If  a  family  with  an  income  of  $2200  a  year  spends 
16%  of  its  income  for  rent  and  26%  for  food,  what  amount 
does  it  spend  for  each  of  these  items  ? 

Students  should   be   encouraged  to  prepare  family  budgets  at 
home,  with  the  help  of  their  parents. 


28  ARITHMETIC  OF  THE  HOME 

Exercise  19.    Household  Economics 

1.  A  grocer  sells  coffee  in  half-pound  packages  at  19  $ 
a  package  and  in  4-pound  cans  at  $1.40  a  can.   If  a  woman 
wishes  4  lb.,  what  per  cent  does  she  save  in  purchasing 
by  the  can  ? 

2.  If  you  can  buy  Dutch  cocoa  in  ^-pound  boxes  at  24$ 
a  box  or  in  4-pound  cans  at  $2.65  a  can,  and  you  wish  4  lb., 
what  per  cent  do  you  save  in  purchasing  by  the  can  ? 

3.  If  you  can  buy  maple  sirup  at  480  a  quart  or  in  gallon 
cans  at  $1.50  a  can,  and  you  wish  1  gal.  of  sirup,  what  per 
cent  do  you  save  in  purchasing  by  the  can  ? 

4.  If  a  woman  can  buy  corn  at  15  $  a  can  or  $1.50  per 
dozen  cans,  what  per  cent  does  she  save  on  4  doz.  cans  in 
buying  by  the  dozen  ? 

5.  If  a  woman  can  buy  soup  at  20$  a  can  or  $2.10  per 
dozen  cans,  what  per  cent  does  she  save  on  2  doz.  cans  in 
buying  by  the  dozen  ? 

6.  A  woman  can  buy  a  bushel  of  potatoes  for  80$  or  a 
peck  for  25$.    If  she  needs  a  bushel  of  potatoes,  what  per 
cent  does  she  save  if  she  buys  by  the  bushel  ? 

7.  A  woman  can  buy  -|-  doz.  cans  of  tomatoes  for  75  $ 
or  1  can  for  15$.    If  she  needs  ^  doz.  cans,  what  per  cent 
does  she  save  if  she  buys  by  the  half  dozen? 

8.  If  flour  costs  $7.40  a  barrel  (196  lb.)  or  5$  a  pound, 
what  per  cent  does  a  family  save  in  purchasing  flour  by 
the  barrel  if  it  requires  196  lb.  ? 

9.  If  a  family's  ice  bill  averages  $1.75  a  month,  and  ice 
costs  35$  per  100  lb.,  how  many  pounds  does  the  family 
use  ?    If  by  having  a  better  ice  box  there  is  a  saving  of 
10%  in  the  amount  of  ice  used,  how  many  pounds  are 
used  ?    How  much  is  now  the  average  ice  bill  per  month  ? 


HOUSEHOLD  ECONOMICS  29 

10.  If  you  can  buy  some  chairs  for  $24  cash  or  $3  down 
and  $3  a  month  for  8  mo.,  what  per  cent  do  you  save  if 
you  pay  cash  ? 

In  Exs.  10-12  interest  is  not  to  be  considered  at  this  time.  It 
should  be  mentioned  incidentally  as  a  subject  to  be  studied  later. 

11.  If  you  can  buy  a  sewing  machine  for  $40  cash  or 
by  paying  $4  a  month  for  a  year,  what  per  cent  do  you 
save  if  you  pay  cash  ? 

12.  If  a  reduction  of  10%   is  allowed  on  all  electric- 
light  bills  paid  before  the  tenth  of  each  month,  what  amount 
would  be  saved  in  4  mo.  if  advantage  is  taken  of  this  rule 
in  the  account  on  page  23  ? 

13.  After  the  holidays  the  price  of  toys  in  a  certain  store 
was  reduced  40%.    How  much  would  you  save  by  waiting 
until  after  the  holidays  to  buy  a  mechanical  toy  that  was 
marked  $3.50  before  the  reduction? 

In  the  following  problems  use  the  current  market  price  as 
found  by  inquiry  at  home  or  at  the  store : 

14.  Find  the  per  cent  which  you  can  save  in  purchasing 
each  of  the  following  in  5-pound  packages  instead  of  by 
the  pound:  sugar,  starch,  prunes,  raisins. 

The  teacher  may  omit  Exs.  14-16  if  desired.  A  few  such  problems, 
in  which  the  students  supply  the  data,  serve,  however,  to  make  the 
subject  more  real. 

15.  Find  the  per  cent  saved  in  purchasing  each  of  the 
following  by  the  dozen  cans :  tomatoes,  corn,  peaches. 

As  a  matter  of  economy  it  should  be  noticed  that  it  is  not  always 
good  policy  to  purchase  in  large  amounts  because  the  material  may 
deteriorate  or  be  wasted. 

16.  Find  the  per  cent  saved  in  purchasing  potatoes  by 
the  bushel  instead  of  by  the  peck. 


30 


ARITHMETIC  OF  THE  HOME 


Exercise  20.    Miscellaneous  Problems 

1.  Mr.  Anderson  earns  $28  a  week.    He  spends  20%  of 
his  income  for  rent,  26%  for  food  for  the  family,  6%  for 
fuel  and  lights,  18%  for  clothing  for  the  family,  10%  for 
church  and  charity,  and  2%  for  incidentals.    How  much  is 
left  each  year  for  other  expenses  and  for  savings  ? 

Although  1  yr.  =  52^-  wk.,  or  52f  wk.   in  leap  years,  52  wk.  is 
always  to  be  taken  as  a  year  in  problems  of  this  type. 

2.  The  girls  in  a  class  in  millinery  need   20  yd.   of  a 
certain  quality  of  ribbon.     They  can  buy  this  ribbon  at 
220  a  yard,  or  5yd.  for  $1.    What  per  cent  will  be  saved 
if  they  take  the  latter  plan  ? 

3.  The  goods  for  a  certain   dress  cost   $7.80  and  the 
buttons  and  trimmings  cost  $2.20.     The  cost  of  making 
the  dress  is  60%  of  the  cost  of  the  materials.    If  a  dress 
of  like  quality  and  style  can  be  bought  for  $15,  what  per 
cent  is  saved  by  buying  the  dress  ready  made  ? 

4.  Make  out  a  blank  like  the  one  shown  below,  but 
extended  to  include  your  entire  school  program,  and  com- 
pute the  per  cent  of  time  devoted  to  each  subject: 


TIME  OF 
RECITATION 

SUBJECT 

MINUTES  OF 
RECITATION 

PER  CENT 
OF  TOTAL 

MINUTES 
OF  STUDY 

PERCENT 
OF  TOTAL 

9-935 
940-1015 

Arithmetic 
English 

35 
35 

5.  Some  girls  made  30  pieces  of  candy  from  the  follow- 
ing recipe:  3  cups  granulated  sugar,  150;  1^-  cups  milk,  30; 
^  cake  chocolate,  2^-0;  an  inch  cube  of  butter,  20.  The 
fuel  cost  them  20,  and  they  sold  the  candy  at  the  rate  of 
3  pieces  for  50.  What  per  cent  was  gained  on  the  cost? 


EEVIEW  31 

Exercise  21.   Review  Drill 
Add,  and  also  subtract,  the  following : 

1.  2.  3.  4. 

$750.68      $680.01      $630.27      5  ft.  4  in. 
298.98       297.56       429.68      2  ft.  6  in. 


Multiply  the  following  : 

5.                            6.  7.                              8. 
$298.63               $342.80  $674.39  2ft.  7  in. 
27               92  129  8_ 

Divide  as  indicated,  to  two  decimal  places : 
9.  $426.34  H-  7.        10.  3469.1 -=-16.     ••  11.  427jn-0.6. 

12.  $275  is  what  per  cent  more  than  $200  ? 

13.  $200  is  what  per  cent  less  than  $275?    what  per 
cent  less  than  $300  ? 

14.  If  17  cars  cost  $13,600,  how  much  will  9  cars  cost? 

15.  How  much  is  72%  of  350  lb.?    of  $3500?   of  3£? 

16.  How  much  is  175%  of  $2500?  12%  of  $625?  £%  of 
$2000?  |%  of  $1200?  11%  of  $2400? 

Write  the  answers  to  the  following : 

17.  39,987  +  46,296.  22.  CDXL  =  (?> 

18.  73,203-59,827.  23.  321  ft.  -  (?)  yd. 

19.  34J  x  $42,346.  24.  18£  gal.  -  (?)  qt. 

20.  429. 75 -s- 25.  25.  4|bu.  =  (?)pk. 

21.  J  of  25  ft.  4  in.  26.  16  mL=  (?)  yd. 

In  all  such  drill  work  the  teacher  should  keep  a  record  of  the  time 
required  by  the  students  to  solve  the  problems.  Each  student  should 
strive  to  improve  his  record  when  reviewing  the  page  later  in  the  year. 


32  ARITHMETIC  OF  THE  HOME 

Exercise  22.    Problems  without  Numbers 

1.  If  you  have  an  account  with  several  items  of  income 
and  several  items   of  expenses,  how  do  you  proceed  to 
balance  the  account  ? 

2.  How  would  you  proceed  to  make  out  a  household 
account  for  a  week  ? 

3.  How  do  you  find  what  per  cent  of  the  week's  income 
is  spent  for  household  expenses  ? 

4.  If  you  know  the  income  of  a  household  and  know 
what  per  cent  of  the  income  is  allowed  for  food,  how  do 
you  find  the  amount  allowed  for  food  ? 

5.  If  you  know  what  fraction  of  his  income   a  man 
spends  for  rent,  how  do  you  find  what  per  cent  he  spends 
for  rent? 

6.  If  you  wish  to  know  before  you  receive  the  gas  bill 
the  amount  of  gas  consumed  at  your  home  next  month, 
how  will  you  proceed  to  read  the  meter  ? 

7.  If  you  know  the  cost  of  tomatoes  per  dozen  cans 
and  the  cost  per  can,  how  will  you  find  the  per  cent  of 
saving  of  a  person  who  purchases  a  dozen  cans  at  the 
dozen  rate  instead  of  by  the  can  ? 

8.  If  you  know  the  recipe  for  making  cake  for  a  certain 
number  of  persons,  how  will  you  change  the  recipe  if  you 
are  making  enough  for  a  certain  other  number  of  persons? 

9.  If  a  man  wishes  a  set  of  dining-room  furniture  and 
finds  that,  by  waiting  a  week,  he  can  buy  it  at  a  mark- 
down  sale  at  a  certain  rate  per  cent  off.  the  regular  price, 
how  will  you  find  the  amount  he  will  save  by  waiting? 

10.  If  you  know  how  much  a  man  paid  for  rent  last 
year  and  how  much  more  he  pays  this  year,  how  will  you 
find  the  per  cent  of  increase  ? 


ARITHMETIC  OF  THE  STORE  33 

II.   ARITHMETIC  OF  THE  STORE 

Nature  of  the  Work.  Fred  Dodge  applied  for  a  position 
in  a  store.  The  manager  asked  him  if  he  could  add  a 
column  of  figures  quickly  and  correctly,  and  if  he  could 
compute  quickly  in  his  head.  Fred  thought  he  could,  but 
when  the  manager  tested  him  it  was  found  that  Fred  was 
lacking  in  two  things:  he  had  not  been  taught  to  check 
his  work,  and  he  did  not  know  the  common  short  cuts 
in  figuring  that  are  used  in  all  stores. 

Fred  found  that  the  arithmetic  work  which  he  needed 
most  was  addition,  making  change,  and  multiplication. 
We  shall  briefly  review  these  subjects. 

In  this  review  special  attention  will  be  given  only  to  such  topics 
as  are  not  generally  treated  in  the  elementary  arithmetic  which 
precedes  this  course. 

Oral  Addition.  In  adding  two  numbers  like  48  and  26 
mentally,  it  is  better  to  begin  at  the  left.  Simply  think 
of  68  (which  is  48  +  20)  and  6,  the  sum  being  74.  This 
is  the  way  the  clerk  in  the  store  adds  48$  and  26$. 

Exercise  23.   Addition 

All  work  .oral 

Add  the  following,  beginning  at  the  left  and  stating  only 
the  results  : 

1.  2.  3.  4.  5.  6. 

29      68      75      38      45      68 
15      23      21      24      25      27 

7.       8.       9.       10.      11.      12. 

75      76      88      95      75      80 
25      26      75      25      30      75 


34  ARITHMETIC  OF  THE  STORE 

Exercise  24.   Addition 

See  how  long  it  takes  to  copy  and  add  these  numbers,  check- 
ing the  additions  and  writing  the  total  time : 


1. 

2. 

3. 

4. 

4287.75 

4349.08 

4476.82 

4495.53 

425.90 

346.58 

345.46 

228.69 

381.92 

238.46 

38.69 

642.95 

360.58 

190.84 

248.60 

72.38 

5. 

6. 

7. 

8. 

4458.65 

4394.38 

4298.05 

4482.60 

97.86 

26.95 

342.60 

234.65 

230.95 

700.00 

20.07 

381.90 

48.38 

83.56 

78.68 

32.83 

621.04 

75.08 

.  380.95 

300.00 

9. 

10. 

11. 

12. 

42080.60 

43064.45 

44148.20 

43275.25 

3679.70 

817.66 

876.42 

842.35 

909.36 

4239.58 

3192.68 

3065.06 

517.38 

86.38 

2124.45 

2095.05 

2310.25 

3098.07 

629.00 

812.40 

•  3096.65 

2901.94 

5082.00 

3028.60 

13. 

14. 

15. 

16. 

46240.45 

44063.45 

43083.95 

42438.65 

239.76 

398.43 

498.76 

480.00 

3865.42 

2200.75 

298.80 

3557.76 

396.37 

4346.68 

3763.84 

463.48 

900.48 

328.93 

.  2989.95 

3086.75 

1637.00 

4043.68 

4263.49 

2945.50 

ADDITION  AND  SUBTRACTION  35 

Oral  Subtraction.  In  subtracting  mentally  it  is  better  to 
begin  at  the  left  except  in  making  change.  In  the  case 
of  52-28  think  simply  of  32  (which  is  52-20)  and  take 
8  from  it,  leaving  24. 

This  subtraction  may  be  treated  by  the  process  of  making  change, 
next  described.  Students  should  be  familiar  with  both  processes. 

Making  Change.  If  you  owe  640  to  a  merchant  and 
give  him  $1,  he  says,  "  64  and  1  are  65,  and  10  are  75, 
and  25  are  $  1,"  or,  briefly,  "  64,  65,  75,  $1,"  at  the  same 
time  laying  down  10,  100,  and  250. 

The  merchant  will  first  lay  down  the  coin  or  coins  that  will  bring 
the  amount  up  to  a  multiple  of  5  ;  then  the  largest  coin  or  coins  that 
will  bring  it  up  to  a  multiple  of  25  ;  and  so  on. 

Exercise  25.    Subtraction 

All  ivork  oral 
Subtract  the  following  numbers : 

1.  2.  3.  4.  5.  6.  7. 

47    47    47    56    73    83    95 
30    33    39    28    34    36    48 

Imagine  yourself  selling  goods  at  a  store  and  receiving  in 
each  case  the  first  amount  given,  the  goods  costing  the  second 
amount.  State  the  amount  of  change  due  in  each  case,  and 
state  what  coins  and  bills  you  would  give  in  change  : 

8.  $1;  840.           11.  $3;  $2.20.         14.  $5;  $2.65. 

9.  $2;  $1.25.       12.  $4;  $3.56.         15.  $10;  $7.75. 
10.  $2;  $1.78.        13.  $5;  $2.28.         16.  $5;  $2.35. 

The  teacher  should  ask  the  students  to  find  how  a  cash  drawer  is 
arranged,  and  should  describe  the  cash  register.  A  little  work  in 
making  change  with  real  or  toy  money  may  profitably  be  given. 


36  ARITHMETIC  OF  THE  STORE 

Exercise  26.  Subtraction 

See  how  long  it  takes  to  copy  these  numbers,  to  subtract,  and 
to  check  by  adding  each  difference  to  its  subtrahend;  ivrite 
the  total  time  with  the  last  result : 

1.  2.  3.  4. 

74,856  24,965  44,430  34,008 

36,278  18,986  36,898  30,975 

5.  6.  7.  8. 

75,500  38,990  78,006  60,900 

34,965  29,009  38,869  36,969 

9.  10.  11.  12. 

$275.68  $220.85  $308.06  $600.04 

46.99  165.90  88.79  189.86 

13.  14.  15.  16. 

$278.00  $470.41  $309.20  $202.70 

149.96  82.64  67.64  32.96 

17.  18.  19.  20. 

$402.64  $300.00  $408.72  $472.92 

89.85  183.75  45.86  88.96 

21.  22.  23.  24. 

$309.92  $482.60  $300.00  $425.30 

43.48  193.84  285.68  226.98 

25.  26.  27.  28. 

$329.80  $408.73  $496.05  $506.00 

49.96  229.84  309.78  329.80 


MULTIPLICATION  37 

Oral  Multiplication.  When  Fred  went  to  work  in  the 
store  he  found  that  he  often  needed  to  multiply  rapidly. 
For  example,  if  he  sold  7  yd.  of  cloth  at  45$  a  yard,  he 
needed  to  find  the  total  selling  price  at  once,  without  using 
a  pencil.  lie  found  that  it  was  usually  easier  to  begin 
at  the  left  to  multiply.  In  the  case  of  7  x  45$  he  simply 
thought  of  7  X  40$,  or  $2.80,  and  35$,  making  $3.15  in  all. 

Exercise  27.   Multiplication 

Examples  1  to  12,  oral 

Multiply  the  following,  beginning  at  the  left :. 
1.  2.  3.  4.  5.  6. 

45  38  32  56  56  65 

_6  _4  _7  _8  _9  _7 

7.  8.  9.  10.  11.  12. 

72  77  56  45  55  35 

_§          -1          —          _§          _Z          _§ 

Multiply  the  following : 

13.  43  x  473.  21.  355  x  926.  29.  35  x  6464. 

14.  38  x  308.  22.  280  x  628.  30.  42  x  8480. 

15.  29  x  247.  23.  84  x  6088.  31.  68  x  9078. 

16.  66  x  385.  24.  29  x  4756.  32.  39  x  4030. 

17.  425  x  736.  25.  63  x  2798.  33.  203  x  3405. 

18.  520  x  826.  26.  42  x  4802.  34.  330  x  4143. 

19.  332  x  509.  27.  34  x  3006.  35.  223  x  6062. 

20.  477  x  805.  28.  23  x  3989.  36.  447  x  3095. 

Teachers  who  care  to  give  the  check  of  casting  out  nines  may  do 
so  at  this  time.  Algebra  is  required,  however,  for  its  explanation. 


238174. 


38  ARITHMETIC  OF  THE  STORE 

Short  Cuts  in  Multiplication.  You  have  already  learned 
in  arithmetic  that  there  are  certain  short  cuts  in  multipli- 
cation. These  short  cuts  can  be  used  extensively  in  the 
store.  The  most  important  ones  are  as  follows : 

To  multiply  by  10,  move  the  decimal  point  one  place  to  the 
right ;  annex  a  zero  if  necessary. 

To  multiply  by  100  or  1000,  move  the  decimal  point  to  the 
right  two  or  three  places  respectively  ;  annex  zeros  if  necessary. 

To  multiply  by  3,  multiply  by  10  and  divide  by  2. 

To  multiply  by  25,  multiply  by  100  and  divide  by  4. 

To  multiply  by  125,  multiply  by  1000  and  divide  by  8. 

To  multiply  by  33^,  multiply  by  100  and  divide  by  3. 

To  multiply  by  9,  multiply  by  10  and  subtract  the  multi- 
plicand. 

To  multiply  by  11,  multiply  by  10  and  add  the  multiplicand. 

Exercise  28.    Short  Cuts 
Find  the  results  mentally  whenever  you  can 
Multiply,  in  turn,  by  10,  by  100,  by  5,  by  25,  and  by  125 : 

1.  6456  9248  25,192         23,848         22,200 

2.  8168  9.376         19,920.8         25.088         58.752 

3.  5776         24.432         56,246.4         23.048         46.832 

Multiply,  in  turn,  by  33 j,  by  9,  and  by  11 : 

4.  46,977         67,053         15,240         17,604         13,806 

5.  441.54         466.74         1639.2         457.05         96.816 

6.  483.66         190.56         1804.5         20.166         69.306 

Multiply,  in  turn,  by  5,  by  25,  and  by  50 : 

7.  15,384         56,812         87,824         756,52         73.728 

8.  86,988         47,752         93,104         527.24         43.332 


SHOKT  CUTS  IN  MULTIPLICATION  39 

Multiply  the  following : 

9.  10  x  10.35.  20.  331  x  45.  31:  1000  x  $7.62. 

10.  10  x  $225.  21.  33£  x  288.  32.  50  x  $4220. 

11.  10  x  $7.75.  22.  33J  x  585.  33.  125  x  $3200. 

12.  50  x  $652.  23.  100  x  $45.  34.  331  x  $345. 

13.  50  x  $345.  24.  100  x  $33.  35.  16|  x  186. 

14.  25  x  $544.  25.  1000  x  $65.  36.  16|  x  $696. 

15.  25  x  $280.  26.  25  x  $85.35.  37.  5  x  40,364. 

16.  25  x  $428.  27.  12|  x  $4400.  38.  5  x  $15,680. 

17.  675  x  $35.  28.  12|  x  $4088.  39.  125  x  $408. 

18.  25  x  $5.20.  29.  125  x  $560.  40.  125  x  $4.08. 

19.  10  x  $4.80.  30.  125  x  $5600.  41.  16|  x  7200. 

42.  How  much  will  25  books  cost  at  80$  each? 

43.  How  much  will  25  yd.  of  cloth  cost  at  16$  a  yard  ? 

44.  How  much  will  50  cans  of  corn  cost  at  14$  each? 

45.  How  much  will  4  books  cost  at  75$  each? 

46.  How  much  will  25  yd.  of  cloth  cost  at  24$  a  yard  ? 

47.  How  much  will  12^  yd.  of  cloth  cost  at  48$  a  yard  ? 

48.  How  much  will  80  doz.  pencils  cost  at  56$  a  dozen  ? 

49.  How  much  will  75  books  cost  at  60$  each? 

50.  How  much  will  75  coats  cost  at  $5  each  ? 

51.  How  much  will  a  man  earn  in  48  wk.  at  $25  a  week  ? 

52.  How  will  3^-  doz.  cans  of  tomatoes  cost  at  12$  a  can  ? 

53.  At  $7.50  each,  how  much  will  11  tables  cost? 

54.  At  $8.25  each,  how  much  will  9  desks  cost? 

55.  At  $9.60  each,  how  much  will  25  chairs  cost? 

56.  At  $42.50  each,  how  much  will  11  typewriters  cost? 


40  ARITHMETIC  OF  THE  STORE 

Product  of  an  Integer  and  a  Fraction.  In  the  store  we 
frequently  have  to  find  the  product  of  an  integer  and  a 
fraction.  For  example,  we  may  need  to  find  the  cost  of 
:f  yd.  of  velvet  at  $2  a  yard.  As  we  have  learned, 

To  find  the  product  of  a  fraction  and  an  integer,  multiply 
the  numerator  of  the  fraction  by  the  integer  and  write  the 
product  over  the  denominator. 

Before  actually  multiplying,  indicate  the  multiplication  and 
cancel  common  factors  if  possible. 

Reduce  the  result  to  an  integer,  a  mixed  number,  or  a 
common  fraction  in  lowest  terms. 

For  example,  to  multiply  -g-|-  by  18.  Since  we  have  -g-|> 
if  we  have  18  times  as  much  we  shall  have 


22 

or  T*  °r 


While  we  use  ^  as  an  illustration,  we  seldom  find  in  a  store  any 
need  for  a  common  fraction  with  a  denominator  larger  than  8. 

Exercise  29.    Multiplication 

All  work  oral 
Multiply  the  following,  using  cancellation  when  possible  : 


1. 

-|  of  6. 

6. 

48  x 

¥• 

11. 

jL  of  48. 

16. 

4  of 

960. 

2. 

f  of  8. 

7. 

50  x 

9 

T^' 

12. 

24  x 

7 
•8- 

17. 

fof 

272. 

3. 

48 

xf. 

8. 

40  x 

3^. 

13. 

128  x  f  . 

18. 

864 

x  -J. 

4. 

80' 

x|. 

9. 

*of 

35. 

14. 

132) 

<iV 

19. 

484 

x  -3. 
x  4. 

5. 

50 

x* 

10. 

fof 

88. 

15. 

*of 

36. 

20. 

192 

x4 

x  8. 

Multiplication  of  a  fraction  by  a  fraction  is  not  so  frequently 
needed  in  the  store  as  the  work  given  above. 


MULTIPLICATION  41 

Multiplication  involving  Mixed  Numbers.  In  the  store 
we  frequently  need  to  find  such  a  product  as  12^  x  16,  as 
in  reckoning  the  cost  of  12^- yd.  of  cloth  at  160  a  yard. 
In  this  particular  example  we  should  simply  think  of  1g°' 
of  16,  or  200,  and  state  the  result  at  once  as  |2.  But  in 
general,  as  we  have  already  learned, 

To  multiply  a  mixed  number  by  an  integer,  multiply 
separately  the  integral  and  fractional  parts  of  the  mixed 
number  by  the  integer  and  add  the  products. 

For  example,  7  x  2|  =  14^  =  19 J. 

To  multiply  a  fraction  by  a  fraction,  multiply  the  numer- 
ators together  for^  the  numerator  of  the  product  and  the  de- 
nominators together  for  the  denominator  of  the  product. 

This  case,  familiar  to  the  student,  is  mentioned  here  for  the  sake 
of  completeness. 

To  multiply  a  mixed  number  by  a  mixed  number,  reduce 
each  to  an  improper  fraction  and  multiply  the  results,  using 
cancellation  whenever  possible. 

For  example,  2  J  x  5f  =  |  x  -^  =  1J£  =  14|. 


Exercise  30.   Multiplication 

Multiply  the  following : 

1.  15  x  3f      8.  9|  x  320.      15.  31  x  25J. 

2.  18  x  2|.      9.  9|  x  688.      16.  12|  x  15f . 

3.  36  x  5f.      10.  35f-  x  18.      17.  25f  x  34|. 

4.  80  x  9|.      11.  17f  x  65.      18.  14|  x  16-f. 


5.  48  x  7i.  12.  261  x  84.  19.  18|  x  231. 

£                                 A  TC             O 

6.  36J  x  48.  13.  25J  x  320.  20.  181  x  27}. 

7.  321  x  45.  14.  37.  x  57..  21.  15J  x  28f . 


42  ARITHMETIC  OF  THE  STORE 

Use  of  Aliquot  Parts  in  Multiplication.  As  you  have  al- 
ready learned  and  would  naturally  infer  from  page  38  and 
from  a  few  examples  already  met,  it  is  easier  to  multiply 
$£,  $J,  and  $J  than  it  is  to  multiply  12J#,  16f  <£,  and  33-^. 
Such  parts  of  a  dollar  or  of  any  other  unit  are  often  called, 
as  we  have  learned,  aliquot  parts. 

At  33^$  each,  15  books  cost  15  times  $^,  or  $5. 

At  16|0  each,  24  rulers  cost  24  times  $J,  or  $4. 

At  12-|$  each,  16  notebooks  cost  16  times  $-|,  or  $2. 

While  goods  are  seldom  marked  16|-$  or  $-|,  they  are 
often  marked  6  for  $1,  which  is  the  same  thing. 

Exercise  31.   Aliquot  Parts 

All  work  oral 

1.  At  12^0  a  yard,  how  much  will  32  yd.  of  cloth  cost  ? 

2.  At  16^  a  yard,  how  much  will  36  yd.  of  cloth  cost  ? 

3.  At  33^0  a  yard,  how  much  will  39  yd.  of  cloth  cost  ? 

4.  At  8  notebooks  for  $1,  how  much  will  24  notebooks 
cost  ?    How  much  will  32  notebooks  cost  ? 

5.  At  16-|$  each,  find  the  cost  of  42  glass  vases. 

6.  At  the  rate  of  6  pairs  for  $1,  how  much  will  48 
pairs  of  scissors  cost? 

7.  How  many  children's  coats  can  be  cut  from  7|-  yd. 
of  cloth,  allowing  2^  yd.  to  a  coat  ? 

State  the  products  of  the  following : 

8.9x33^.  12.  24xl6f£.  16.16x12^. 

9.18x33^.          13.  66xl6f<£.  17.  48  x 

10.  27  x  33J^.          14.  48  x  16§<£.  18.  72  x  12 

11.  150x33^.        15.  72xl6f£.  19.  96x12^. 


MULTIPLICATION 


43 


Cash  Checks.    In  many  of  the  large  stores  the  clerks 
are  required  to  fill  out  caslf  checks  like  the  following: 


Sold  by  No.  29    Amount  rec'd,  $4-0.00 

2±  yd. 

Velvet 

/.SO 

3 

75 

/¥§   " 

£/atiAt 

.80 

// 

50 

/6i   "  • 

&fc 

.<?5 

/5 

68 

Total  

30 

33 

Change  due  

07 

W 

00 

Exercise  32.   Cash  Checks 

Make  out  cash  checks  for  the  following  sales: 

1.  3J  yd.  cotton  @  180,  24  yd.  velveteen.  @  87J0, 16Jyd. 
dimity  @  300.    Amount  received,  $30. 

2.  8£  yd.  gingham  @  300,  8-^yd.  madras  @  380,  9^  yd. 
silk  @  $1.25.    Amount  received,  $20. 

3.  24Jyd.  linen  @  380,  22 J  yd.  linen  suiting  @  850, 
4^  yd.  dimity  @  280.    Amount  received,  $30. 

4.  6^  yd.  India  linen  @  420,  8J  yd.  cheviot  @  $1.35, 
18  yd.  cotton  @  12^-0.    Amount  received,  $20. 

5.  14  yd.  muslin  @  250,  |-yd.  velvet  @  $3,  6-|yd.  India 
linen  @  450,  12|-yd.  lining  @  110.    Amount  received,  $10. 

6.  24J  yd.  muslin  @  240,  3£  yd.  velvet  @  $2.40,  26|  yd. 
lining  @  120,  6|  yd.  silk  @  $1.60,  5|  yd.  suiting  @  800, 
6^  yd.  ribbon  @  300.    Amount  received,  $50. 

The  teacher  should  ask  the  students  to  make  problems  similar  to 
those  given  above,  using  the  local  prices  of  common  materials. 


44  ARITHMETIC  OF  THE  STOEE 

Discount.  When  goods  are  sold  at  less  than  the  marked 
price,  the  reduction  in  price  fe  called  discount. 

Local  examples  should  be  mentioned  and  the  students  should  be 
asked  for  illustrations  of  discount  within  their  experience. 

List  Price.  The  price  of  goods  as  given  in  a  printed 
catalogue  or  list  issued  by  the  manufacturer  or  by  the 
wholesale  house  is  called  the  list  price.  In  stores  where 
the  goods  are  marked,  this  is  called  the  -marked  price. 

Discount  is  usually  reckoned  as  a  certain  per  cent  or 
as  a  certain  common  fraction  of  the  list  price  or  marked 

price,  thus:   20%  off,  33^%  off,  -|  off,  and  so  on. 

i 
Net  Price.     The  price  of  goods  after  the  discount  has 

been  taken  off  is  called  the  net  price. 

Cash  Discount.  A  discount  allowed  because  the  pur- 
chaser pays  at  once  is  called  a  cash  discount. 

For  example,  a  Boy  Scout  suit  may  be  marked  $6,  but 
owing  to  the  desire  of  the  dealer  to  clear  out  his  stock  at 
the  end  of  the  season  he  may  mark  it  to  sell  for  10%  off 
for  cash.  The  suit  will  then  be  marked  |6  less  10%  of  $6, 
or  |6 -$0.60,  or  $5.40. 

Trade  Discount.  When  merchants,  jobbers,  or  manufac- 
turers sell  to  dealers  they  often  deduct  a  certain  amount 
from  the  list  price.  This  reduction  is  called  a  trade  discount. 

Such  terms  as  retail  dealer,  wholesale  dealer  or  jobber,  and  manufac- 
turer should  be  explained  by  the  teacher  if  necessary. 

A  special  form  of  trade  discount  is  allowed  for  very  large  orders. 
This  is  called  a  quantity  discount. 

The  terms  of  discount  are  often  stated  thus :  2/10,  1/30, 
jV/60,  these  symbols  meaning  2%  discount  if  the  bill  is  paid 
'within  10  da.,  1%  if  paid  within  30  da.,  net  (no  discount) 
thereafter,  and  the  bill  to  be  paid  within  60  da. 


DISCOUNTS  45 

Exercise  33.    Discounts 

Examples  1  to  15,  oral 

1.  If  some  goods  are  marked  $20,  and  10%  discount 
is  allowed,  what  is  the  selling  price  ? 

2.  If  a  book  marked  80<£  is  sold  at  a  discount  of  25%, 
this  is  how  many  cents  below  the  marked  price  ? 

3.  If  a  merchant  buys   $800  worth  of  goods  and  is 
allowed  10%  discount  in  case  he  pays  for  them  at  once, 
how  much  does  he  save  by  prompt  payment  ? 

Find  the  discounts  on  the  following  at  the  rates  specified : 

4.  |80,  10%.  8.  $40,  25%.  12.  $120,  10%. 

5.  $25,  10%.  9.  $88,  25%.  13.  $150,  20%. 

6.  $50,  20%.  10.  $60,  50%.  14.  $160,  25%. 

7.  $25,  20%.  11.  $90,  50%.  15.  $250,  50%. 

16.  If  goods  marked  $475   are   sold  to  a   dealer  at  a 
discount  of  20%,  how  much  do  they  cost  him? 

17.  If  a  merchant  marks  a  lot  of  suits  at  $24.75  each, 
and  sells  them  at  -^  off,  what  is  the  net  price  of  each  ? 

Students  should  be  asked  to  watch  advertisements  in  the  news- 
papers to  see  what  discounts  are  offered  and  should 'state  the  prob- 
able reasons  for  these  discounts. 

Griven  the  marked  prices  and  rates  of  discount  as  follows, 
find  the  net  prices : 

18.  $17.50,  10%.  23.  $21.60,  12£%.  28.  $48.60,  25%. 

19.  $16.50,  20%.  24.  $72.30,  33^%.  29.  $27.70,  50%. 

20.  $27.75,  20%.  25.  $38.70,  33^%.  30.  $35.00,  10%. 

21.  $64.40,  25%.  26.  $43.50,  16f  %.  31.  $34.75,  20%-. 

22.  $86.60,  25%.  27.  $86.40,  16|%.  32.  $65.25,  20%. 


46  ARITHMETIC  OF  THE  STORE 

Several  Discounts.  In  some  kinds  of  business  two  or 
more  discounts  are  frequently  allowed.  For  example,  a 
dealer  may  buy  hardware  listed  at  $200  with  discounts 
of  20%,  10%  (20%  and  10%,  often  called  and  written 
simply  20,  10).  This  means  that  20%  is  first  deducted 
from  the  list  price,  and  then  10%  from  the  remainder. 

The  list  price  is  $200. 

The  list  price  less  20%  is  $160. 

Then  $160  less  10%  is  $144,  the  net  price. 

In  reckoning  discounts  the  student  should  discard  every  fraction 
of  a  cent  in  the  several  discounts. 

Exercise  34.   Several  Discounts 
Examples  1  to  5,  oral 

1.  From  $100  take  10%,  and  5%  from  the  remainder. 

2.  From  $800  take  25%,  and  1%  from  the  remainder. 

3.  From  $600  take  20%,  and  20%  from  the  remainder. 

4.  A  dealer  bought  some  goods  at  a  list  price  of  $100, 
with  discounts  of  10%,  10%.    How  much  did  he  pay? 

Find  the  net  prices  of  goods  marked  and  discounted  as 
follows : 

5.  $400,  20%,  30%.  9.  $1300,  15%,  10%. 

6.  $1400,  35%,  5%.  10.  $800,  20%,  10%. 

7.  $800,  20%,  15%.  11.  $600,  15%,  10%. 

8.  $1200,  12%,  4%.  12.  $550,  10%,  10%. 

13.  What  is  the  difference  between  a  discount  of  50% 
on  $1000,  and  the  two  discounts  of  25%,  25%  ? 

14.  Is  there  any  difference  between  a  discount  of  5%, 
4%,  and  one  of  4%,  5%,  on  $900  ?    How  is  it  on  $600  ? 


SEVERAL  DISCOUNTS  47 

Sample  Price  List.  The  following  is  a  price  list  of 
certain  school  supplies,  with  the  discounts  allowed  to 
schools  and  dealers  when  the  prices  are  not  net: 

Composition  books,  $4.80  per  gross,  less  5% 

Drawing  compasses,  1.75  per  doz.,  less  10%,  5% 
Drawing  paper,  9  x  12,         1.30  per  package,  less  10% 

Drawing  pencils,  4.80  per  gross,  less  20% 

Penholders,  3.40  per  gross,  less  12%,  4% 

Pens,  0.65  per  gross,  less  25%,  10% 

Rulers,  0.40  per  doz.,  net 

Thumb  tacks,  0.45  per  100,  less  30% 

Tubes  of  paste,  4.15  per  gross,  less  10%,  6% 

Exercise  35.    Purchases  for  the  School 

1.  A  school  board  wishes  to  buy  8  packages  of  drawing 
paper  and  200  thumb  tacks.    How  much  will  they  cost  ? 

In  this  exercise  use  the  above'  price  list. 

2.  How  much  will  12  gross  of  pens  and  2  gross  of  com- 
position books  cost  ?  4  gross  of  pens  and  3  doz.  rulers  ? 

3.  There  are  18  students  in  a  class,  and  each  student 
needs  compasses  and  a  ruler.     How  much  will  all  these 
drawing  instruments  cost  the  school  ? 

4.  If  a  dealer  sells  pens  at  a  cent  apiece,  how  much 
does  he  gain  per  gross  ?    If  he  sells  penholders  at  3  §  each, 
how  much  does  he  gain  per  gross? 

5.  A  dealer   buys    3   gross    of   rulers    and   2   gross   of 
drawing  pencils.     He   sells  the  rulers  and  pencils  at  5^ 
each.    How  much  does  he  gain  in  all  ? 

6.  If  a  dealer  buys  2  gross  of  tubes  of  paste  for  mount- 
ing pictures  and  sells  the  tubes  at  5$  each,  how  much 
does  he  gain  on  the  purchase  ?  how  much  per  gross  ? 


48 


ARITHMETIC  OF  THE  STORE 


Bill  with  Several  Discounts.    The  following  is  a  common 
form  of  a  jobber's  bill  of  goods  with  several  discounts : 


Burlington,   la.,   TWfMf  13,  19  SO 


Mv. 

Vougnt  of  RQBERTS  g,  STONE,  Jewelers 

1072   Passaic  Avenue 
Terras:  20°fo,   /O% 


/O 


8  cloy,  afawtz-  @  //¥.  7o 

7  day.  fotatzd  ptifa,  @    $£.20 

Lew,  20%,  /0°jo 


1)8 


00 


f06 


27 


Exercise  36.   Bills 

Make  out  bills  for  the  following : 

1.  4  doz.  sweaters  at  $34.    Discounts  10%,  5%. 

2.  16  doz.  files  at  $7.30.    Discounts  25%,  20%. 

3.  625  yd.  taffeta  at  fl.35;   240  yd.  velvet  at  $1.80. 
Discounts  15%,  10%. 

4.  6  doz.  pairs  hinges  at  $4.50 ;  12  doz.  table  knives  at 
$9.20.    Discounts  20%,  10%. 

5.  16  doz.  locks  at  $4.50  ;  4  doz.  mortise  locks  at  $4.85. 
Discounts  20%,  8%. 

6.  840  yd.  taffeta  at  $1.10 ;  12  gross  pompons  at  $150 ; 
4  doz.  pieces  braid  at  $21.60.    Discounts  10%,  5%. 

7.  960yd.  silk  at  $1.75;  640yd.  lawn  at  270;   860yd. 
taffeta  at  $1.05.    Discounts  10%,  5%,  5%. 


49 


Receipted  Bill.     The  following  is  a  receipted    bill  for 
some  goods  purchased  by  a  retail  merchant  from  a  jobber : 


Mi,,  ft.  TW. 
'Bought  of 


rft 

Ter 


Chicago,  111,   &&«,.   /7, 

,  fctUt,  M. 


STARR  &  TIFFANY,  Jewelers 

8378  Burlington  Ave. 


66 


68 


4-0 


In  this  case  Mr.  Nourse  is  the  debtor  (Dr.),  since  he  is  in 
debt  for  the  amount ;  the  firm  of  Starr  &  Tiffany  is  the 
creditor  (Cr.),  since  it  trusts  Mr.  Nourse,  or  gives  him 
credit.  On  this  bill  only  a  single  discount  was  allowed. 

A  receipt  may  also  be  written  separately  instead  of  appearing  on  a 
bill.  Such  a  receipt  should  be  dated,  and  should  be  in  substantially 

this  form:  "Received  from the  sum  of dollars  for ." 

The  receipt  should  be  signed  by  the  creditor. 

The  subject  of  commercial  discount  is  of  great  value  because  of  its 
extensive  use  not  only  in  wholesale  transactions  but  even  in  bargain 
sales.  Students  should  understand  that  some  of  the  reasons  for  allow- 
ing discounts  are  buying  in  large  quantities  and  paying  cash  down 
or  within  a  specified  time,  and  they  should  become  familiar  with  the 
ordinary  deductions  from  list  prices.  It  is  well,  for  obvious  reasons, 
to  consider  bills  and  receipts,  preferably  of  a  local  character,  in  con- 
nection with  the  study  of  this  topic. 


50 


ARITHMETIC  OF  THE  STORE 


Invoice.    A  bill  stating  in  detail  a  list  of  items  and  prices 
of  goods  sold  is  called  an  invoice.   A  sample  invoice  follows  : 


St.  Paul,  Minn., 


1920 


'Bought  of 


BROTHERS 

IMPORTERS  OF  DRY  GOODS  AND  FANCY  GOODS 


236 


2,  W,  4-f,  4-2, 


,  y-O 


676  33 


3?, 


V-OS 


In  the  above  invoice  411  means  41^;  412  means 
or  41^;  and  413  means  41|^.  This  is  the  customary  way 
of  indicating  the  number  of  yards  in  pieces  of  goods.  The 
numbers  236  and  427  refer  to  the  price  list  in  which  the 
goods  are  described.  The  numbers  14  and  8  indicate  the 
number  of  pieces  (pc.)  bought.  The  number  of  yards  of 
the  first  is  576  and  of  the  second  324. 

The  expression  "  Terms :  10  da."  means  that  Mr.  Dunbar 
has  10  da.  in  which  to  pay  for  the  goods.  Such  an  invoice 
may  or  may  not  mention  the  discount  allowed. 

There  is  no  essential  arithmetic  difference  between  a 
bill  of  goods  and  an  invoice  of  a  wholesale  dealer. 


INVOICES  51 

Exercise  37.    Invoices 

Make  out  invoices  for  the  folloiving : 

1.  24  doz.  caps  @  $15.50,  2J  doz.  ties  @  $8.40. 

2.  32  pc.  ribbon  @   $1.25,  14  pc.    @   $1.30,  48  pc.   @ 
$1.75,  24  pc.  @  $1.12|,  32  pc.  @  $1.37^,  48  pc.  @  $1.25, 
36  pc.  @  $1.40,  64  pc.  @  $1.60. 

3.  3  carloads  coal,  21,700  lb.,  24,200  lb.,  25,100  lb.,  @ 
$5.20  per  short  ton ;  3  carloads  coal,  22,700  lb.,  21,900  lb., 
20,400  lb.,  @  $5.75  per  short  ton.    Terms:  4%  for  cash. 

4.  1  gro.  cans   sardines   @  $3.40  per  doz.,  9  doz.  cans 
shrimps   @   $1.75,  8  doz.  tins  herrings  @   $2.50,  6^  doz. 
cans  lobster  @  $2.88,  32  cans  mackerel  @  16-|^,  4-|  doz. 
cans  kippered  herrings  @  $2.65.    Terms:  4%  for  cash. 

5.  8  doz.  packages  codfish  @  $1.80,  9  doz.  cans  salmon  @ 
$2.40,  15  doz.  cans  caviar  @  $3.50,  18  doz.  cans  Yarmouth 
bloaters  @  $2.40,   24  doz.  cans  tongue  @  $8.75,   16  doz. 
cans  baby  mackerel  @  $1.80.    Terms:   3^%  for  cash. 

6.  12  doz.   jars  meat   extract   @    $3.40,,    24  doz.   cans 
chicken  @  $3.15,  24  doz.  cans  beef  @  $2.40,  18  doz.  cans 
soup  @  $3.40,  9  doz.  cans  clam  chowder  @  $3.75,  16  doz. 
cans  clam  juice  @  $1.10.    Terms:   3%,  2%. 

7.  8  armchairs  @   $6.75,    24  kitchen    chairs  @   $1.25, 
12  kitchen  tables  @  $2.25,    6  bedroom  sets  @  $42.50,  24 
rockers  @  $8.25,  16  dining  tables  @  $12.50,  9  sideboards 
@  $16.66f.    Terms:    6%,  4%. 

8.  20  pc.  linen  containing  401,  402,  383,  42,  411,  402,  40, 
40,  401,  412,  43,  393,  401,  421,  402,  401,  413,  42,  391,  40  yd., 
@   75^;  16  pc.  silk  containing  411,  421,  40,  392,  42,  403, 
38,  411,  43,  413,  39,  402,  401,  411,  40,  392yd.,  @  $1.40. 
Terms:  6%,  3%. 


52  ARITHMETIC  OF  THE  STORE 

Exercise  38.    Review 

1.  Make  out  an  invoice  for  the  following  goods  purchased 
May  7  and  paid  for  May  10,  terms  2/10,  1/20,  N/W : 
10  bolts  dress  linen,  10,  12,  II1,  122,  103,  II3,  12,  101,  II2, 
II3 yd.,  @  480  a  yard;  2*4  bolts  French  nainsook,  1 2  yd.  each, 
@  18$  a  yard;  30  bolts  mercerized  lingerie  batiste,  24  yd. 
each,  @  22$  a  yard;  32  bolts  imported  lawn,  10  yd.  each, 
@  460  a  yard.    Insert  names  and  find  the  net  cost. 

2.  In  Ex.  1  find  the  net  cost  if  the  payment  is  made 
on  May  20. 

3.  Make  out  an  invoice  for  the  following  goods  purchased 
Sept.  20  and  paid  for  Oct.  5,  terms  2/20,  JV/90 :  5  No.  264 
plows  @  $42.50  less  20%;  3  No.  178  self-dump  hayrakes 
@  $18.60  less  15%;  9  No.  325  hay  stackers  @  $46.50  less 
15%.    Insert  names  and  find  the  net  cost. 

4.  A  jobber  offers  his  customers  discounts  of  15%,  10%, 
but  the  invoice  clerk  made  a  mistake  on  a  bill  of  $85  and 
gave  a  single  discount  of  25%.    How  much  did  the  error 
cost  the  clerk  or  the  jobber  ?  t 

5.  A  manufacturer  lists  a  desk  at  $52  less  25%,  and  a 
rival  manufacturer  offers  a  similar  desk  for  $57  less  ^.    If 
the  first  dealer  increases  his  discount  to  25%,  3%,  which 
will  be  the  lower  net  price  and  how  much  lower? 

6.  A  retail  dealer  being  allowed  a  discount  of  20%,  2%, 
find  the  net  price  of  the  following  goods  purchased  from  a 
wholesale  dealer  at  the  list  prices  stated :  1  dining  table, 
$34 ;  8  chairs  @  $2.85  ;  1  buffet,  $32.50  ;  1  rug,  $26. 

7.  A  dressmaker  bought  the  following  bill  of  goods, 
receiving  a  trade  discount  of  8%  and  a  cash  discount  of 
5%:  8Jyd.  broadcloth  @  $3;  13£yd.  silk  lining  @  80 0; 
3  yd.  trimming  @  $2.40.    What  was  the  net  price  ? 


REVIEW  53 

Exercise  39.   Review  Drill 

1.  Multiply  488  by  £;  by  0.5;  by  50%  ;  by  50. 

2.  Multiply  24  by  33J%  ;  by  25%  ;  by  12|%. 

3.  Multiply  |  by  | ;  f  by  f ;  f  by  f . 

4.  Divide  1  by  £;  J  by  J;  f  by  J;  J  by  f. 

5.  Multiply  8  by  J;  8  by  0.125;  8  by  125. 

6.  Divide  4800  by  100 ;  by  10 ;  by  20 ;  by  40. 

7.  Add  |-  and  ^;  ^  and  ^;  |-  and  £. 

8.  Find  the  values  of  1-f ;  IJ-f;  5J  -  3| . 

9.  Express  as  decimals  and  also  as  common  fractions: 

;  16f%;  62£% ;  66f  % ;  87^%;  83£% ;  33J%. 

10.  Express  12  ft.  as  inches ;   96  in.  as  feet ;  69  ft.  as 
yards ;  51  yd.  as  feet ;  16  gal.  as  quarts ;  16  qt.  as  gallons  ; 
3  Ib.  4  oz.  as  ounces ;  32  oz.  as  pounds. 

11.  |500  is  what  per  cent  of  $400? 

12.  $500  is  what  per  cent  more  than  $400  ?  than  $250  ? 

13.  $400  is  what  per  cent  less  than  $500  ?  than  $800  ? 

14.  If  8  typewriters  cost  $500,  how  much  will  15  cost? 

15.  If  9  desks  cost  $63,  how  much  will  36  desks  cost? 

16.  If  the  rent  of  an  apartment  is  $720  for  ^yr.,  how 
much  is  the  rent  for  a  year  ? 

Perform  the  following  operations : 

17.  41f  in.  +  ljin.  22.   25%  of  $1.60. 

18.  5|in.-2|in.  23.  $2.75-7-l|. 

19.  2|  x  27J  in.  24.  §  of  34J  ft. 

20.  6 J  ft.  -T-  3^  ft.  25.  J  of  9  ft.  4  in. 

21.  3.75-S-2J.  26.  75%  of  6  Ib.  4  oz. 


54  AKITHMETIC  OF  THE  STORE 

Exercise  40.    Problems  without  Numbers 

1.  If  you  buy  a  fishing  rod  and  sell  it  at  a  certain  rate 
per  cent  above  cost,  how  do  you  find  the  selling  price  ? 

2.  If  a  man  buys  a  car  and  sells  it  at  a  certain  per 
cent  below  cost,  and  you  know  how  much  he  paid  to  keep 
the  car  and  how  much  he  saved  by  using  it,  how  do  you 
find  whether  he  gained  or  lost,  and  the  amount  ? 

3.  If  you  know  the  number  of  school  days  in  the  year 
and  the  number  of  times  you  were  absent  from  school  dur- 
ing the  year,  how  do  you  find  the  percentage  of  absences  ? 

4.  If  you  know  the  number  of  times  and  the  percentage 
of  times  a  train  arrives  on ,  time  during  a  certain  month, 
how  do  you  find  the  number  of  runs  the  train  makes? 

5.  If  you  sell  a  person  a  certain  bill  of  goods,  and  he 
hands  you  more  than  the  required  amount  of  money,  how 
do  you  proceed  to  make  change  ? 

6.  If  you  know  a  man's  income  and  the  amount  which 
he  pays  for  rent,  how  do  you  find  what  per  cent  of  his 
income  he  pays  for  rent? 

7.  If  you  know  the  rate  and  amount  of  an  agent's 
commission,  how  do  you  find  the  selling  price  ? 

8.  How  do  you  express  a  given  per  cent  as  a  decimal  ? 
as  a  common  fraction  ? 

9.  If  you  know  what  a  certain  per  cent  of  a  number 
is,  how  do  you  find  the  number? 

10.  How  do  you  find  what  per  cent  one  given  number 
is  of  another  given  number? 

11.  If  you  sell  a  man  a  number  of  articles  at  a  certain 
price  each,  and  this  price  is  an  aliquot  part  of  a  dollar, 
what  is  the  shortest  method  of  finding  the  amount  due  ? 


ARITHMETIC  OF  THE  FARM  55 

III.   ARITHMETIC  OF  THE  FARM 

Nature  of  the  Work.  The  farming  industry  is  one  of  the 
largest  in  our  country.  There  are  between  six  and  seven 
million  farms  in  the  United  States  and  their  total  area  is 
nearly  900,000,000  acres.  These  farms  are  worth,  with 
their  buildings  and  machinery,  over  $40,000,000,000  and 
they  produce  about  $10,000,000,000  annually.  From  these 
immense  sums  it  will  be  seen  how  important  is  the  farm- 
ing industry  and  how  necessary  it  is  to  know  some  of  the 
problems  relating  to  it. 

Every  boy  and  every  girl  who  lives  on  a  farm  ought  to 
know  how  to  measure  a  field  and  find  its  area,  how  to  keep 
farm  accounts,  and  how  to  make  the  necessary  computa- 
tions relating  to  the  dairy,  the  crops,  and  the  soil.  Even 
boys  and  girls  who  live  in  villages  and  cities  should, 
for  their  general  information,  know  something  of  these 
matters,  just  as  those  who  live  on  the  farm  should  know 
something  about  the  problems  of  the  city. 

The  work  of  measuring  land  and  computing  such  volumes  as  the 
farmer  uses  is  taken  up  in  the  geometry  in  this  book,  but  the  famil- 
iar case  of  the  area  of  a  rectangle  is  assumed  to  be  understood. 

Teachers  in  village  and  city  schools  will  find  in  the  following 
pages  a  sufficient  number  of  problems  for  their  purposes,  but  the 
entire  topic  may  be  omitted  if  necessary.  In  rural  schools,  however, 
additional  problems  should  be  drawn  from  the  locality  in  which  each 
school  is  placed.  Agricultural  products,  the  soils,  the  customs,  the 
wages,  and  the  prices  all  vary  greatly  in  different  sections  of  our 
country,  and  the  teacher  should  encourage  the  students  to  bring 
to  school  problems  which  relate  to  local  interests.  Problems  about 
irrigation  are  important  in  some  states,  while  in  others  they  are 
quite  unheard  of ;  the  alfalfa  crop  is  very  important  in  certain  parts 
of  the  country,  while  in  others  it  is  not ;  land  is  laid  out  in  sections 
in  some  states,  while  it  is  never  so  laid  out  in  others.  The  teacher 
should  be  guided  by  a  knowledge  of  these  various  customs. 


56  AEITHMETIC  OF  THE  FAKM 

Exercise  41.    Cost  of  Wastefulness 

1.  If  a  farm  wagon  that  cost  |60  is  left  out  in  the  yard 
instead  of  being  kept  in  the  shed,  it  will  last  about  6  yr., 
but  if  kept  under  cover,  it  will  last  about  twice  as  long. 
What  per  cent  of  the  cost,  not  considering  the  interest  on 
the  money,  does  a  farmer  pay  for  his  carelessness  per  year 
if  he  leaves  the  wagon  out  of  doors? 

2.  A  farmer  after  threshing  his  wheat  had   16  T.   of 
straw  left.    A  ton  of  this  straw  contains  10  Ib.  of  nitrogen 
worth  150  a  pound,  18  Ib.  of  potassium  worth  6^  a  pound, 
and  2  Ib.  of  phosphorus  worth  1 2  ^  a  pound.    If  the  farmer 
wastes  the  straw  instead  of  using  it  on  the  soil  as  fertilizer, 
how  much  money  does  he  waste  ? 

3.  It  is  computed  that  a  certain  kind  of  farm  machinery 
depreciates  in  value  as  follows,  if  reasonably  good  care  is 
taken  of  it:  10%  of  the  original  value  the  first  year,  8%  of 
the  original  value  the  second  year,  6  %  of  the  original  value 
the  third  year,  3%  of  the  original  value  the  fourth  year 
and  each  year  thereafter.    A  machine  of  this  kind  cost  a 
farmer  $240.    He  did  not  take  proper  care  of  it,  and  at 
the  end  of  4  yr.  it  was  worth  only  $115.    The  per  cent  of 
the  original  value  thus  lost  in  the  4  yr.  was  how  much 
more  than  the  per  cent  that  would  have  been  lost  had 
proper  care  been  taken  of  the  machine  ? 

4.  The  strip  of  waste  land  along  each  side  of  a  woven - 
wire  fence  is  2  ft.  6  in.  wide,  the  strip  along  a  barbed-wire 
fence  is  3  ft.  wide,  and  the  strip  along  a  rail  fence  is  4  ft. 
6  in.  wide.    How  many  rods  of  each  kind  of  fence  cause 
a  waste  of  1  acre  of  land  on  one  side  ?    At  $90  an  acre, 
find  the  value  of  the  land  wasted  along  one  side  of  80  rd. 
of  each  kind  of  fence ;  along  one  side  of  200  rd.  of  each 
kind  of  fence. 


FARM  ACCOUNTS 


57 


Farm  Accounts.  Many  careful  farmers  keep  systematic 
accounts  of  the  receipts  and  expenditures  for  each  of  their 
fields,  as  well  as  for  the  farm  as  a  whole.  In  the  problems 
of  the  following  exercise  an  itemized  statement  is  given  of 
expenditures  for  a  20-acre  field  of  corn. 

Exercise  42.    Farm  Accounts 
1.  In  the  following  account  supply  the  missing  amounts: 


EXPENSES 

Apr. 

9 

Plowing,  51  da.                 @  $4.80 

11 

Harrowing,  2-|-  da.            @  $4.50 

29 

2bu.  seed  corn                  @  $1.00 

30 

Planting,  2  da.                   @  $4.80 

May 

16 

Replanting,  1  da.               @  $1.25 

17 

Harrowing,  2  da.               @  $4.50 

23 

Plowing,  4  da.                   @  $4.75 

June 

9 

Plowing,  3  da.                   @  $4.75 

28 

Plowing,  3  da.                   @  $4.75 

July 

15 

Plowing,  3  da.                   @  $4.75 

Sept. 

19 

Cutting,  6  da.                    ©  $4.00 

Nov. 

8 

Husking,  718  bu.              @     4<£ 

20 

Rent  of  land  @  $5.50  per  acre 

Total  

2.  If  the  receipts  in  the  above  case  came  from  the  sale 
of  718  bu.  of  corn  at  62$,  with  $40  worth  kept  on  hand, 
the  expenditures  are  what  per  cent  of  these  receipts? 

3.  In  the  field  of  Exs.  1  and  2  find  the  net  profit. 

In  rural  schools  it  is  desirable  to  secure  or  to  have  the  students 
secure  local  accounts  of  this  kind.  This  is  the  best  way  to  make  the 
subject  real  to  those  who  are  studying  it. 


58 


ARITHMETIC  OF  THE  FARM 


Exercise  43.    Farm  Records 

1.  Ralph's  father  explained  to  him  what  was  meant  by 
grading  the  cows,  keeping  their  records  for  milk,  and  sell- 
ing the  poor  cows.  He  showed  him  this  farm  record: 


WITH  SYSTEMATIC  GRADING 

WITHOUT  SYSTEMATIC  GRADING 

Herd 

Annual  cost 
of  food 
per  cow 

Annual 
profit 
per  cow 

Herd 

Annual  cost 
of  food 
per  cow 

Annual 
profit 
per  cow 

a 
b 

$34.28 
33.37 

$34.02 
36.19 

9 
h 

131.65 
40.58 

$13.34 
8.13 

c 

47.11 

25.83 

i 

38.30 

22.22 

d 

e 

36.72 
31.19 

38.27 
48.86 

j 
k 

37.40 
33.78 

22.33 
38.93 

f 

31.57 

42.81 

I 

32.88 

14.91 

Ralph  and  his  father  then  figured  out  the  average  annual 
cost  of  food  and  profit  per  cow,  in  each  class  of  herds. 
They  did  this  by  dividing  by  6  the  total  of  each  of  the 
four  columns.  What  were  the  results? 

2.  At  a  certain  experiment  station  the  five  most  profit- 
able and  the  five  least  profitable  cows  compared  as  follows : 


COST  OF 

POUNDS  OF 

AVERAGE 

GRADE  OF  Cows 

FEED 
AND  CARE 

BUTTER 

FAT 

COST  OF 

1  LB.   OF 

ANNUALLY 

ANNUALLY 

BUTTER  FAT 

Five  most  profitable  cows 

156.54 

304 

Five  least  profitable  cows 

52.02 

189 

Compute  the  average  cost  of  1  Ib.  of  butter  fat  for  the 
most  profitable  and  for  the  least  profitable  cows. 


PROBLEMS  OF  THE  DAIRY  59 

Exercise  44.    Problems  of  the  Dairy 

1.  A  farmer  sells  26,250  Ib.  of  milk  to  a  creamery  in  a 
certain   month.     The   milk   averages  4.2%   by  weight  of 
butter  fat.    With  how  many  pounds  of  butter  fat  does  the 
creamery  credit  the  farmer  in  that  month  ? 

2.  If  6  Ib.  of  butter  fat  are  needed  in  making  7  Ib.  of 
butter,  what   is  the  value  of  the  butter  produced  from 
1236  Ib.  of  butter  fat,  the  butter  being  worth  34$  a  pound  ? 

3.  A  certain  dairy  sells  to  a  creamery  milk  averaging 
3.75%  of  butter  fat.    The  butter  fat  weighs  630  Ib.    How 
many  pounds  of  milk  does  the  dairy  sell  ? 

4.  A  farmer  has  two  cows,  one  supplying  986  Ib.  of  milk 
testing  3.1%  butter  fat  in  a  certain  month,  and  the  other 
812  Ib.  testing  4.2%.    If  the  creamery  allows  the  farmer 
32$  a  pound  for  butter  fat,  which  cow  pays  him  the  more 
and  how  much  more,  the  feed  and  care  costing  the  same  ? 

5.  A  creamery  uses  7500  Ib.  of  milk  in  a  week.    The 
skim  milk  amounts  to  80%  of  the  whole  milk  and  con- 
tains 0.6%  butter  fat.     How  many  pounds  of  butter  fat 
are  lost  in  the  skim  milk  ? 

6.  A  farmer  owns  a  herd  of  18  cows  that  average  25  Ib. 
of  milk  per  head  daily.    This  milk  tests  3.5%  butter  fat, 
and  the  butter  fat  is  worth  26.5$  per  pound.    How  much 
does  the  farmer  receive  in  30  da.  for  the  butter  fat  ? 

7.  A  herd  of  24  cows  averages  22  Ib.  of  milk  per  cow 
daily,  and  another  herd  of  18  cows  averages  28  Ib.  per  cow. 
The  milk  of  the  first  herd  averages  5%  butter  fat  and  that 
of  the   second  herd  3.5%.     How  many  more  pounds  of 
butter  fat  are  produced  by  the  first  herd  per  week  ? 

In  rural  communities  there  should  be  special  computations  on 
such  subjects  as  rations,  cost  of  labor,  and  the  income  from  cows. 


60 


ARITHMETIC  OF  THE  FARM 


8.  Given  the  following  table  showing  the  number  of 
pounds  of  nitrogen,  phosphorus,  and  potassium  in  1000  Ib. 
of  each  of  five  kinds  of  feed,  find  the  per  cent  of  each  of 
these  three  ingredients  in  each  of  the  five  kinds: 


FEED 

NITROGEN 

PHOSPHORUS 

POTASSIUM 

Wheat  straw    .     . 

5.0 

0.8 

9.0 

Timothy  hay     .     . 
Clover  hay  .     .     . 
Corn  

12.0 
20.0 
17.1 

1.5 
2.5 
3.0 

11.8 
15.0 
3.4 

Wheat    .     . 

23.7 

4.0 

4.3 

9.  The  following  table  shows  the  amount  of  protein 
and  carbohydrates  in  certain  kinds  of  feed: 


WEIGHT  OF  A 

POUNDS  PI 

B  BUSHEL 

POUNDS 

Protein 

Carbohydrates 

%e 

56 

5.0 

39 

Barley     .     .     . 
Corn  

48 
56 

4.0 
3.5 

32 
40 

Oats  

32 

3.0 

19 

A  dairy  cow  of  average  size  requires  daily  about  2  Ib.  of 
protein  and  12  Ib.  of  carbohydrates.  When  corn  is  62$  and 
oats  41$  per  bushel,  which  is  the  cheaper  food,  considering 
the  protein  alone  ?  considering  carbohydrates  alone  ? 

10.  In  Ex.  9  the  weight  of  the  protein  in  a  bushel  of  rye 
is  what  per  cent  of  the  weight  of 'the  rye?  Answer  the 
same  question  for  barley ;  for  corn ;  for  oats.  Answer 
the  same  questions  for  the  carbohydrates. 

The  weight  of  a  bushel  yari.es  in  the  different  states. 


FEEDING  CORN  61 

Exercise  45.    Feeding  Corn 

1.  When  corn  was  selling  at  550  a  bushel,  a  farmer  de- 
cided to  feed  his  corn  to  his  cattle.    He  estimated  that  the 
increase  in  the  value  of  the  cattle,  from  the  corn  alone,  was 
60<£  for  each  bushel  used  for  feed.    What  was  the  per  cent 
of  increase  in  the  value  of  the  corn  by  using  it  as  feed  ? 

2.  A  record  of  the  result  of  feeding  corn  to  hogs  was 
kept  on  several  farms.    On  one  farm,  when  corn  was  sell- 
ing at  550  a  bushel,  it  was  found  that  the  increase  in  the, 
value  of  the  hogs  was  equivalent  to  820  per  bushel  of 
corn  fed  to  hogs.    What  was  the  per  cent  of  increase  in 
the  value  of  the  corn  by  using  it  as  feed  ? 

3.  On  another  farm  the  figures  of  Ex.  2  were  520  a 
bushel  for  corn  when  sold  and  750  a  bushel  when  used 
as  feed.    What  was  the  per  cent  of  increase  in  the  value 
of  the  corn  by  using  it  as  feed  ? 

4.  A  bushel  of  corn  contains  about  -|  Ib.  of  nitrogen,  ^  Ib. 
of  phosphorus,  and  ^  Ib.  of  potassium.    How  many  pounds 
of  each  of  these  substances  are  contained  in  the  crop  from 
a  20-acre  field  yielding  62  bu.  of  corn  to  the  acre? 

5.  If  it  costs  $11.80  per  acre  to  grow  a  crop  of  corn 
and  haul  it  to  the  market,  where  it  is  sold  at  600  per 
bushel,  what  is  the  net  profit  from  a  25-acre  field  if  the 
rent  on  the  land  is  $9  per  acre  and  the  land  yields  an 
average  of  58  bu.  of  corn  per  acre  ? 

6.  A  cow  is  fed  daily  6.5  Ib.  of  corn  worth  600  per 
bushel  of  56  Ib.    What  will  be  the  cost  of  the  corn  fed 
to  the  cow  in  2  mo.  of  30  da.  each  ? 

7.  If  a  bushel  of  corn,  when  fed  to  hogs,  wiU  produce 
9.5  Ib.  of  pork,  how  much  will  it  cost  to  produce  1  Ib.  of 
pork  when  corn  is  620  a  bushel? 


62  ARITHMETIC  OF  THE  FARM 

Exercise  46.    Farm  Income 

1.  A  farmer  owns  two  farms  of  the  same  size  and  value. 
One  he  runs  himself  and  the  other  he  lets  on  shares.    The 
following  table  shows  the  itemized  income  of  each  farm : 

HOME  FAKM  RENTED  FARM 

Dairy  products |348.60  $103.75 

Wool 43.75  28.80 

Eggs  and  poultry    ....     316.80  123.50 

Domestic  animals     ....     637.50  321.60 

Crops 1072.80  785.30 

Find  the  total  receipts  on  each  farm. 

2.  In  Ex.  1  find  the  per  cent  of  increase  of  each  item 
of  income  on  the  home  farm  over  the  corresponding  item 
on  the  rented  farm. 

3.  The  following  table  shows  the  itemized  expenses  of 
the  two  farms  mentioned  in  Ex.  1 : 

HOME  FARM  RENTED  FARM 

Labor $212.60  $140.30 

Fertilizers 92.30  12.00 

Feed 63.50  58.60 

Maintaining  buildings  .     .     .       31.75  63.40 

Maintaining  equipment     .     .       15.50  42.50 

Taxes  and  miscellaneous  .     .       82.50  80.75 

Find  the  total  expenses  of  each  farm,  the  difference  in 
each  pair  of  items,  the  difference  in  the  total  expenses, 
and  the  net  gain  of  each  farm. 

4.  In  Ex.  3  find  what  per  cent  more  was  paid  for  labor 
on  the  home  farm  than  on  the  rented  farm.  • 

5.  In   Ex.  3   find   what   per  cent   more    was    paid   for 
maintaining   buildings   on  the  rented   farm  than   on   the 
home  farm. 


SOILS  AND  CROPS  63 

Exercise  47.   Soils  and  Crops 

1.  The  soil  of  an  acre  of  rich  land  in  the  Corn  Belt, 
plowed   to   the    depth   of    6|-  in.,    is   estimated    to   weigh 
2,000,000  Ib.  and  to  contain  8000  Ib.  of  nitrogen,  2000  Ib. 
of  phosphorus,  35,000  Ib.  of  potassium,  and  1ST.  of  cal- 
cium carbonate  (limestone).   Express  each  of  these  weights 
as  per  cent  of  the  total  weight. 

The  figures  given  in  the  problems  on  this  page  are,  as  usual  in 
such  cases,  only  approximations,  because  soil  and  crops  vary  greatly 
in  different  places.  The  figures  are,  however,  always  based  upon 
scientific  results  as  obtained  in  agricultural  experiment  stations. 

2.  The  plant  food  liberated  from  the  soil  during  an 
average  season  is  2%   of  the  nitrogen,  1%   of  the  phos- 
phorus, and  ^  %  of  the  potassium  contained  in  the  surface 
stratum  of  6|-  in.  mentioned  in  Ex.  1.    Find  the  number 
of  pounds  of  each   of  these   elements   liberated  at  these 
rates  from  a  100-acre  field  in  1  yr. 

3.  The  grain  in  a  100-bushel  crop  of  corn  takes  from 
the  soil  100  Ib.  of  nitrogen,  17  Ib.  of  phosphorus,  and  19  Ib. 
of  potassium,  and  the  stalks  take  48  Ib.,  6  Ib.,  and  52  Ib. 
respectively.    Express  each  of  the  first  three  as  per  cent  of 
the  total  weight  of  the  corn,  allowing  56  Ib.  to  the  bushel. 
Allowing  60  Ib.  of  stalks  to  produce  1  bu.  of  shelled  corn, 
express  each  of  the  last  three   as  per  cent   of  the  total 
weight  of  the  stalks. 

4.  Given  that  1 T.  of  clover  hay  contains  40  Ib.  of  nitro- 
gen, 5  Ib.  of  phosphorus,  and  30  Ib.  of  potassium,  express 
each  as  per  cent  of  the  total  weight  of  the  hay. 

5.  If  50  bu.  of  wheat,  weighing  60  Ib.  per  bushel,  con- 
tains 12  Ib.  of  phosphorus,   13  Ib.  of  potassium,  4  Ib.  of 
magnesium,  1  Ib.  of  calcium,  and  0.1  Ib.  of  sulphur,  each 
is  what  per  cent  of  the  weight  of  the  wheat  ? 


64  ARITHMETIC  OF  THE  FARM 

Exercise  48.    Good  Roads 

1.  A  teamster  had  to  haul  7^-T.  of  barbed  wire  a  dis- 
tance of  13  mi.  over  poor  roads  from,  the  railway.   He  found 
that  he  could  haul  only  1500  Ib.  to  a  load  and  that  it  took 
him  a  full  day  to  make  the  round  trip.    How  long  did  it 
take  him  to  haul  the  7-|  tons  of  wire,  and  how  much  did 
it  cost  at  $5  per  day  for  man  and  team? 

2.  In  Ex.  1  what  was  the  cost  of  hauling  1  ton  1  mi., 
or  the  cost  of  1  ton-mile,  as  it  is  ordinarily  called  ? 

3.  In  Ex.  1,  after  a  new  state  road  had  been  constructed, 
the  teamster  found  that  with  the  same  team  he  could  haul 
24-  T.  to  the  load  and  make  the  round  trip  in  1  da.    How 
much  did  it  then  cost  to  haul  a  ton  of  wire  ?    What  was 
the  cost  per  ton-mile  ? 

4.  Comparing  the  results  in  Exs.  2  and  3,  what  per 
cent  less  was  the  cost  per  ton-mile  in  Ex.  3,   owing   to 
good  roads?    What  was  the  per  cent  of  time  saved  in 
hauling  7J  T.? 

5.  A  farmer  lives  10  mi.  from  the  railway.     The  road 
was  formerly  so  bad  that  with  a  two-horse  team  he  could 
haul  only  30  bu.  of  wheat  to  the  load,  and  it  took  1  da.  to 
make  the  round  trip.    At  $5  a  day  for  man  and  team,  how 
much  did  it  cost  per  bushel  to  haul  the  wheat  ? 

6.  The  roads  were  recently  improved.    The  farmer  can 
now  haul  75  bu.  to  the  load.    Allowing  ^  da.  for  the  trip, 
find  the  cost  per  bushel  of  hauling  the  wheat  now. 

7.  Comparing  Exs.  5  and  6,  how  much  more  money  due 
to  unproved  roads  does  the  farmer  get  per  bushel  ? 

8.  Taking  60  Ib.  as  the  weight  of  a  bushel  of  wheat 
and  comparing  Exs.  5  and  •  6,  what  has  been  the  per  cent 
of  reduction  in  cost  of  cartage  per  ton-mile  ? 


REVIEW  65 

Exercise  49.    Review  Drill 

1.  Express  0.4%  as  a  decimal  fraction. 

2.  Express  2.8  as  per  cent. 

3.  Express  2|-  qt.  as  per  cent  of  1  gal. ;  of  5  gal. 

4.  Express  an  inch  as  per  cent  of  1  yd. ;  of  3  yd. 

5.  Express  8£%,  16f%,  33J%,  62J%,  66f%,  83J%, 
and  87|-%  as  common  fractions  in  lowest  terms. 

Find  the  value  of  each  of  the  following : 

6.  50%  of  860.  11.  0.7%  of  275  Ib. 

7.  25%  of  62J.  12.  0.08%  of  56,000. 

8.  37J%  of  $9600.  13.  225%  of  4800. 

9.  133^%  of  $19.56.  14.  2.25%  of  4800. 
10.  187^%  of  $19.28.  15.  300%  of  156.7. 

Multiply  as  indicated: 

16.  275  x  3468.     17.  39.6  x  31.78.     18.  0.432  x  687.2. 

Divide  to  two  decimal  places : 

19.  ln-3.7.  20.  0.27-^0.5.         21.  68.01-=-  0.7. 

22.  $30  is  what  per  cent  of  $360  ?  of  $3600  ?  of  $36,000  ? 

23.  8  in.  is  what  per  cent  of  12  in.  ?  of  12  ft.  ?  of  12  yd.  ? 

24.  A  pint  is  what  per  cent  of  1  qt.  ?  of  1  gal.  ?  of  6  gal.  ? 

25.  56  ft.  is  8%  of  what  distance  ?  4%  of  what  distance? 

26.  7ft.  6  in.  is  10%  of  what  distance?    8%  of  what 
distance  ?    33-^%  of  what  distance  ? 

27.  75  Ib.  is  25%  less  than  what  weight? 

28.  A  bill  of  goods  amounting  to  $725   is  allowed  a 
discount  of  15%.    Find  the  net  amount. 


66  ARITHMETIC  OF  THE  FARM 

Exercise  50.    Problems  without  Numbers 

1.  If  you  know  the  expenditures  and  the  receipts  for 
a  year  on  a  farm,  how  do  you  find  the  net  profit  or  loss  ? 

2.  In  Ex.  1  how  do  you  find  the  average  net  profit 
or  loss  per  acre  ? 

3.  If  you  know  the  dimensions  of  a  rectangular  field 
in  rods,  how  do  you  find  the  area  in  square  rods?  in  acres? 

4.  If  you  know  the  total  annual  cost  of  food  for  the 
cows  on  a  farm  and  the  number  of  cows,  how  do  you  find 
the  average  cost  of  food  per  cow  ? 

5.  If  you  know  the  average  profit  per  cow  on  a  farm 
and  the  number  of  cows,  how  do  you  find  the  total  profit? 

6.  If  you  know  the  average  per  cent  of  butter  fat  in 
the  milk  from  a  certain  herd  of  cows  and  the  number  of 
pounds  of  milk  delivered  at  a  creamery,  how  do  you  find 
the  number  of  pounds  of  butter  fat  in  this  milk  ? 

7.  If  a  farmer  has  two  cows,  and  knows  the  amount 
of  milk  furnished  by  each  in  a  year  and  the  per  cent  of 
butter  fat  in  the  milk  of  each  cow,  how  does  he  find  which 
cow  produces  the  more  butter  fat? 

8.  If  you  know  the  weight  of  nitrogen  in  a  ton  of 
clover  hay,  how  do  you  find  the  per  cent  of  the  nitrogen  ? 

9.  If  you  know  the  per  cent  of  nitrogen  in  a  ton  of 
clover  hay,  how  do  you  find  the  weight  of  the  nitrogen  ? 

10.  If  you  know  the  per  cent  of  nitrogen  in  clover  hay, 
how  do  you  find  the  amount  of  clover  hay  necessary  to 
produce  a  given  amount  of  nitrogen  ? 

11.  If  you  know  the  length  of  a  rectangular  field,  how 
do  you  find  the  width  that  must  be  fenced  off  so  as  to 
inclose  just  an  acre  of  land  ? 


AEITHMETIC  OF  INDUSTRY  67 

IV.   AEITHMETIC  OF  INDUSTRY 

Nature  of  the  Work.  We  have  thus  far  considered  three 
important  topics,  the  home,  the  store,  and  the  farm,  and 
have  seen  that  each  has  its  special  kinds  of  problems.  We 
shall  now  consider  the  problems  of  more  general  industries, 
such  as  manufacturing  establishments  of  various  kinds. 

Teachers  should  draw  problem  material  from  local  industries 
whenever  possible.  No  textbook  can  do  more  than  give  a  general 
survey  of  the  subject,  and  it  must  always  endeavor  to  present  prob- 
lems which  are  not  too  technical  to  be  generally  understood.  In 
certain  localities,  however,  where  some  single  industry  is  prominent, 
more  technical  problems  can  safely  be  given  because  the  students 
will  naturally  be  familiar  with  the  terms  used. 

In  order  to  solve  the  problems  which  arise  in  the  shop, 
it  is  necessary  to  review  the  operations  with  numbers.  We 
shall  therefore  briefly  review  the  operations  with  fractions. 

Exercise  51.    Fractions 

1.  Some  plaster  ^  in.  thick  is  coated  with  a  finer  plaster 
YQ  in.  thick.  How  thick  is  the  plaster  then  ? 

3.  A  plate  of  brass  -^  m-  thick  is  laid  on  a  plate  of  iron 
•^  in.  thick.  What  is  the  total  thickness  ? 

3.  An  iron  rod  of  diameter  |-  in.  is  covered  with  a  brass 
plating  ^g-  in.  thick.    What  is  now  the  diameter  of  the  rod  ? 

4.  A  table  4  ft.  4^  in.  long  and  3  ft.  2-|-  in.  wide  has  a 
molding  1  in.   thick   put   around  it.    What   is    then   the 
perimeter  of   the   table  ? 

5.  A  boy  is  making  a  dog  kennel.    One  of  the  pieces 
of  wood  is  4  ft.  3  in.  long,  and  from  this  he  saws  a  piece 
2  ft.  4^  in.  long.     How  long  is  the  piece  which  remains  ? 


68  ARITHMETIC  OF  INDUSTRY 

6.  A  girl  has  a  piece  of  ribbon  2^  yd.  long.    She  uses 
14£  in.  for  a  hat.    How  much  ribbon  has  she  left? 

7.  From  a  board  14  ft.  long  a  man  saws  off  a  piece 
2  ft.  3^  in.  long  and  another  piece  2  ft.  7^  in.  long.    How 
long  is  the  remaining  part  ? 

8.  To  a  piece  of  cloth  4^  yd.  long  another  piece  8£  in. 
long  is  sewed,  and  then  18^  in.  is  cut  off  and  used  for 
making  a  bag.    How  many  yards  of  cloth  are  left  ? 

9.  A  plate   of  glass  18^  in.   by  23  j|  in.  was  set  in  a 
picture  frame  that  covered  it  ^  in.  from  each  edge.    What 
are  the  dimensions  of  the  glass  not  covered  by  the  frame? 

Perform  the  following  additions : 

10.  i  +  f  +  f         14.  i  +  J  +  i-         18.   If +  f  +  3^. 

11.  i+f+f.     15:  J+I  +  &•     19-  2f +1  +  1^. 

12-  i  +  f  +  f.         16.  i  +  l+A-        20.  3f  +  2f+TV 
13.  f +  l  +  f.         17.  f  +  f  +TV        21.  3|  +  21  +  11. 

22.  In  making  a  dress  ruffle  4-|  in.  wide  when  finished 
enough  cloth  must  be  allowed  to  turn  in  ^  in.  on  one  side 

and  -J  in.  on  the  other.    Find  the  width  of  cloth  needed. 

o 

23.  A  gas  fitter,  in  running  a  pipe  into  a  schoolroom, 
has  four  pieces  of  pipe  respectively  8  ft.  9|-  in.,  6  ft.  2-|  in., 
8  ft.  3£  in.,  and  9  ft.  4  in.  long,  and  finds  he  has  3  ft.  7  in. 
more  than  he  needs.    What  is  the  length  required  ? 

Perform  the  following  subtractions : 

24.  51 -2|.  28.   71  -5|.  32.  Sin.  -  2f  in. 

25.  8J -3|.  29.  41 -2|.  33.  9j  in.  -  6^  in. 

26.  61  -2f.  30.  6-4r5g,  34.  8  in.  -  1^  in. 

27.  7| -4}.  31.  5-2jL,  35.  9  in.  -  3^  in. 


EEVIEW  OF  FKACTIONS  69 

Division  by  a  Fraction.   We  know  that  there  are  3  thirds 
in  1,  or  that  1  •*-  -^  =  3.    From  this  we  see  that  6-^-^  =  6  x  3. 

That  is,  6  •*•  £  =  3  x  6, 

and  6  -7-  -i-  =  half  as  much  =  — - —  =  9. 

2 

That  is,  6  -5-  •§-  =  -|-  of  6. 

Therefore,  to  divide  by  a  common  fraction,  multiply  by  the 
reciprocal  of  that  fraction. 

That  is,  15  •*- 1  s=  |  x  IS  =*  25. 

5      3 

3 

35639 
Similarly,  -«.-  =  |x-  =  -. 

4 
We  shall  now  review  both  multiplication  and  division  of  fractions. 


Exercise  52.    Fractions 

1.  A  tin  cup  is  found  to  hold  j-f  pt.    When  it  is  |-  full 
the  cup  contains  what  part  of  a  pint  ? 

2.  If  a  jar  has  a  capacity  of  j-g-  qt.,  what  part  of  a  quart 
will  it  contain  when  it  is  •§•  full?    when  it  is  i-  full? 

o  £ 

What  part  full  must  it  be  to  hold  1  pt.  ? 
Perform  the  following  operations : 


3. 

4 

x  f  . 

9. 

1  x  A. 

15. 

1 
4 

xf 

21. 

4  X 

A- 

4. 

* 

.    3 
•    4* 

10. 

4-*  4 

-i               O 

16. 

1 
4 

.    7 
~  8' 

22. 

i- 

A- 

5. 

a 

.    1 
•    2' 

11. 

i-i 

17. 

7 
¥ 

•*•!• 

23. 

T6-- 

._! 

6. 

i 

xf. 

12. 

ixiV 

18. 

3 
¥ 

xf 

24. 

2x 

T6"' 

7. 

3 

4 

_!_   J. 

.13. 

IV  A- 

19. 

3 

4 

•*'f 

25. 

2-f- 

A- 

8. 

3. 

9 

.    3 
'    4' 

14. 

..!_._._  1 
16    *    2* 

20. 

* 

_=_  3. 

26. 

A- 

4-2. 

AEITHMETIC  OF  INDUSTRY 


Exercise  53.   The  Pay  Roll 
1.  The  following  is  a  week's  pay  roll  of  a  manufacturer : 


PAYROLL                     For  t  Tie  week  ending  Feb  .  7,    1920 

No 

No.  OF  HOURS  PEK  DAY 

TOTAL 

WAGES 

TOTAL 

M. 

T. 

w. 

T. 

;F. 

S. 

TIME 

HOUR 

WAGES 

1 

M.  S.  Rowe 

8 

7* 

8 

6Z 

8 

4 

42 

50 

$21 

2 

T.  D.  Bell 

7 

7£ 

8 

8 

8 

4 

42* 

55 

23 

38 

3 

S.  M.  Lee 

8 

7* 

8 

7£ 

8 

4 

* 

50 

* 

* 

4 

Louis  Shea 

8 

7 

6 

8 

7^ 

3 

* 

55 

* 

* 

5 

C.  P.  Grove 

8 

8 

7 

7* 

8 

3£ 

* 

55 

* 

* 

6 

L.  S.  Cram 

8 

6£ 

8 

0 

7 

3$ 

* 

27£ 

* 

* 

Totals 

* 

# 

* 

* 

* 

* 

* 

* 

* 

Fill  each  space  marked  with  an  asterisk  (*). 

Make  out  pay  rolls  for  a  iveek  and  insert  names,  when  the 
men's  numbers,  the  hours  per  day,  and  the  wages  per  hour 
are  as  follows : 

2.  No.  1:   7|,  7J,  7J,  8,  8,  3|,  600;  No.  2:   8,  8,  8,  8, 
8,  4,  550;  No.  3:  8,  8,  7,  7,  8,  4,  620. 

3.  No.  1:  7,  8,  8,  8,  7£,  4,  62J0;  No.  2:  8,  7,  8,  7,  6, 
4,  600;  No.  3:  8,  7£,  8,  8,  8,  4,  48J0;  No.  4:  8,  8,  8,  8, 
7£,  4,  650;  No.  5:  8,  7,  8,  8,  8,  4,  720. 

4.  No.  1:  8,  7,  6,  8,  8,  4,  72^0;  No.  2:  8,  7,  6,  8|,  8£, 
4,  640;    No.  3:   8,  6,  "6,  8,  8,  4,  620;  No.  4:    7,  8,  7j, 
6£,  8,  4,  570;  No.  5:  8,  6,  7,  7£,  6J,  4,  480. 

5.  No.  1:   7,  7,  8,  8,  8,  4,  62J0;  -No.  2:  8,  8,  7|,  7J,  7J, 
4,  650;  No  3:  8,  8,  8,  7|,  7|,  4,  580;  No.  4:  7J,  7J>  71 
8,  7|,  3|,  620;  No.  5:  8,  8,  8,  8,  8,  4,  63J0. 


THE  PAY  ROLL 


71 


6.  Fill  each  space  marked  with  an  asterisk  in  the  fol- 
lowing pay  roll,  allowing  double  pay  for  all  overtime: 


PAY  ROLL                   For  the  week  ending  Jan  .    10  ,    1920 

No. 

NAME 

No.  OF  HOURS  PER  DAY 

TOTAL 
TIME 

WAGES 

PER 

HOUR 

TOTAL 
WAGES 

M. 

T. 

W. 

T. 

F. 

S. 

1 

R.  S.  Jones 

V/ 

\J 

V/ 

^ 

t 

V/ 

54 

40 

* 

* 

2 
3 

M.  L.  King 
J.  M.  Mead 

V 

x/ 

\J 

V 

I 

I/ 

* 

1 

46 

50 

* 
* 

* 
* 

Totals 

* 

* 

* 

* 

* 

* 

* 

* 

* 

Before  assigning  Ex.  6  the  teacher  should  explain  that  from  one 
and  a  half  to  two  times  the  regular  hourly  wage  is  usually  paid  for 
overtime,  and  that  the  check  (v/)  in  the  above  pay  roll  means  full 
time  for  the  day.  In  this  pay  roll  the  full  time  is  8  hr.  except  on 
Saturday,  when  it  is  4  hr.  The  symbol  %/  means  8  hr.  +  2  hr. 
overtime.  A  dash  ( — )  indicates  absence.  Part  time,  like  6-|  hr., 
is  indicated  as  above  on  Friday  for  King.  Since  the  allowance 
for  overtime  is  double  that  for  regular  work,  Jones's  time  is 
8  +  8  +  8  +  8  +  8  +  4  regular  time  and  2  x  (2  + 1£  +  1|)  overtime, 
or  54  hr.  in  all.  Explain  the  significance  of  the  parentheses. 

Make  out  pay  rolls  (inserting  names')  when  the  men's 
numbers,  the  hours  per  day,  and  the  wages  per  hour  are  as 
follows,  8  hr.  constituting  a  day's  work  except  on  Saturday, 
ivhen  it  is  4  hr.,  and  double  pay  being  given  for  overtime : 

7.  No.  1:  8,  9,  8,  9,  8,  5,  67^0;  No.  2:  8J,  9,  9£,  8,  8, 
4,  650;  No.  3:  8,  8,  8, 10,  8,  6,  62£;  No.  4:  8,  9,  9,  9,  8, 
4,  60^;  No.  5:  8J,  8£,  9,  8,  8,  5,  600. 

8.  No.  1:   8,  10,  8/10,  8,  6,  600;  No.  2:   9J,  8,  8,  9,  8, 
4,  62?  ;  No.  3:  10,  10,  10,  10,  8,  5,  62£0;  No.  4:  8,  8,  8, 
8,  10,  9,  62^0;  No.  5:  8,  8,  8,  9,  8J,  6J,  650. 


72  ARITHMETIC  OF  INDUSTRY 

Exercise  54.    The  Iron  Industry 

1.  What  is  the  weight  of  a  steel  girder  that  is  18'  10'r 
long  and  weighs  46^  Ib.  to  the  running  foot  ? 

2.  What  is  the  cost  of  16'  4"  of  iron  rod,  4^  Ib.  to  the 
foot,  at  !•£$  a  pound? 

3.  The  wooden  pattern  from  which  an  iron  casting  is 
made   weighs   6^%    as   much   as   the   iron.     The   pattern 
weighs  67^-lb.     How  much  does  the  casting  weigh? 

4.  If  steel  rails  weighing  180  Ib.  to  the  yard  are  used 
between  New  York  and  Chicago,  a  distance  of  980  mi., 
how  many  tons   of   rails  will  be  required  for  a  double- 
track  road  between  these  cities? 

5.  An   iron   tire    expands   ^-^Q%    on  being   heated  for 
shrinking  on  a  wheel.    A  certain  wooden  wheel  needs  a 
tire  16' 8"  in  circumference.    How  much  longer  will 'the 
tire   be  when  thus  heated? 

6.  If  3.5%  of  metal  is  lost  in  casting,  how  much  metal 
must  be  melted  to  make  a  casting  to  weigh  77.2  Ib.? 

Since  100%  -  3.5%  =  96.5%,  77.2  is  96.5%  of  the  weight. 

?.•  In  a  certain  blast  furnace  the  casting  machine  turns 
out  40  pigs  of  iron  per  minute,  averaging  in  weight  110  Ib. 
each.  If  this  machine  runs  for  312  da.,  16  hr.  a  day,  how 
many  long  tons  (2240  Ib.)  of  pig  iron  will  it  turn  out  ? 

8.  Some  years  ago  the  average  daily  wages  paid  to  em- 
ployees in  a  certain  mill  was  $1.90,  and  the  men  worked 
11  hr.  a  day,  6  da.  in  the  week.  At  present  the  average 
daily  wage  is  $2.60  and  the  men  work  8  hr.  a  day, 
5  da.  in  the  week  and  5  hr.  on  Saturday.  Considering  the 
wages  per  hour,  what  has  been  the  per  cent  of  increase  :' 
Considering  the  hours  per  dollar  of  wages,  what  has  been 
the  per  cent  of  decrease  ? 


MISCELLANEOUS  PROBLEMS  73 

Exercise  55.    Miscellaneous  Problems 

1.  Sea  Island  cotton  is  usually  shipped  in  bags  of  150  lb., 
while  Alabama  cotton  is  shipped  in  bales  of  500  lb.    How 
many  bags  of  Sea  Island  cotton  at  280  a  pound  will  equal 
in  value  42  bales  of  Alabama  cotton  at  11$  a  pound? 

2.  The  average  number  of  wage  earners  engaged  in  the 
manufacture   of   cotton  goods  during  a  recent  year  was 
379,366.    The  value  of  the  materials  was  $431,602,540 
and  the  value  of  the  finished  products  was  $676,569,335. 
What  per  cent  of  value  was  added  by  manufacture  ? 

3.  The 'United   States   produced    10,102,102   bales    of 
cotton  in  the  year  1900  and  11,068,173  bales  in  the  year 
1915.    What  was  the  per  cent  of  increase  ? 

4.  The  total  value  of  the  cotton  raised  in  the  United 
States  in  a  recent  year  was  $627,861,000,  and  the  number 
of  bales  was  11,191,820.   Find  the  average  value  of  a  bale. 

5.  During  a  recent  year  the  United   States   produced 
11,000,000  bales  of  cotton  and  used  only  7,000,000  bales. 
The  amount  used  in  this  country  was  what  per  cent  of 
the  total  amount  produced  ? 

6.  During  a  recent  year  86,840  sq.  mi.  of  cotton  terri- 
tory was   invaded   by  the    boll  weevil.     The    total    area 

.infected   at   the    end   of   the   year   was   409,014   sq.   mi. 
What  was  the  per  cent  of  increase  for  the  year? 

7.  In  the  days  when  cotton  cloth  was  woven  by  hand 
an  experienced  weaver  could  turn  out  45  yd.  of  cloth  per 
week.    At  present  a  workman  operating  six  power  looms 
in  a  cotton  mill  will  produce  1500  yd.  per  week.     How 
long  would  it  have  taken  the  worker  to  do  this  with  the 
hand  loom?    What  is  the  per  cent  of  increase  in  output 
per  man  with  the  power  looms  ? 


74  ARITHMETIC  OF  INDUSTRY 

8.  Before  the  invention  of  the  cotton  gin  a  laborer  could 
separate  in  a  day  only  1£  Ib.  of  lint  from  the  seed.    At 
the  present  some  gins  turn  out  10  bales  of  500  Ib.  each 
per  day.  Such  a  machine  does  the  work  of  how  many  men  ? 

9.  In  making  a  silk  lamp  shade  the  following  materials 
were  used  :  1J  yd.  silk  @  $1.10,  2J  yd.  silk  fringe  @  $1.84, 
|  yd.  silk.net  @  $2.20, 1  frame  costing  60<£.  The  labor  and 
overhead  charges    amounted  to   $3.25.     The    shade    was 
marked  $14.50  but  was  sold  at  a  discount  of  10%.    Find 
the  gain  per  cent  over  the  total  cost. 

Overhead  charges,  also  called  overhead  or  lurden,  means  the  general 
expense  of  doing  business. 

10.  By  repairing  an  automobile  engine  a  mechanic  in- 
creased its  horse  power  7^%  and  reduced  the  amount  of 
gasoline  necessary  to  run  it  3%.    Before  the  repairs  were 
made  the  engine  developed  40  H.P.  and  used   2  gal.   of 
gasoline  on  a  20-mile  trip.    How  much  gasoline  per  horse 
power  did  it  use  on  a  50-mile  trip  after  it  was  repaired  ? 

The  letters  H.P.  are  commonly  used  for  horse  power. 

11.  It  is  desired  to  construct  an  engine  that  will  generate 
102.5  H.P.  net,  that  is,  actually  available  for  use.    It  is 
found  that  18  %  of  the  horse  power  generated  is  lost.   This 
being  the  case,  what  horse  power  must  be  generated  ? 

12.  How  many  fleeces  of  wool  averaging   6^  Ib.  each 
must  be  used  to  make  a  bale  of  wool  weighing  250  Ib., 
and  how  many  pounds  will  be  left  over? 

13.  If  a  wool  sorter  can  sort  80  Ib.  of  wool  in  a  day, 
how  many  days  will  it  take  him  to  sort  a  shipment  of 
24  bales  of  250  Ib.  each  ? 

14.  After  scouring  (cleaning)  a  shipment  of  12,000  Ib. 
of  wool  it  weighed  only  5240  Ib.     What  per  cent  of  the 
original  weight  was  lost  by  scouring? 


EEVIEW  75- 

Exercise  56.   Review  Drill 

1.  Add  147.832,  29.68,  575,  0.387. 

2.  From  1000  subtract  the  sum  of  148.9  and  9.368. 

3.  Multiply  78.4  by  9.86. 

4.  Divide  0.8  by  0.13  to  three  decimal  places. 

5.  How  much  is  a  profit  of  14^-%  on  a  sale  of  cotton- 
goods  which  cost  $1275.50? 

6.  Find  the  commission  at  ^%  on  goods  sold  for  $15,000. 

7.  Goods  listed  at  $1450  are  sold  at  a  discount  of  6%, 
10%.    Find  the  selling  price. 

8.  How  much  is  the  profit,  at  12^%  on  cost  plus  over- 
head charges,  on  the  sale  of  goods  which  cost  $1645.75, 
the  overhead  charges  being  $268.50  ? 

9.  How  much  is  the  loss  on  a  house  which  cost  $4500, 
including  all  charges,  and  which  was  sold  at  a  loss  of  6  %  ? 

10.  Some  goods  which  cost  $750,  including  all  charges,, 
were  sold  for  $675.    What  was  the  per  cent  of  loss  ? 

Write  the  results  of  the  folloiving : 

11.  84in.  =  (?)ft.  16.  2sq.  ft.  =  (?)sq.  in. 

12.  84oz.  =  (?)lb.  17.  2cu.  ft.  =  (?)cu.  in. 

13.  84ft.  =(?)yd.  18.  2  sq.  yd.  =  (?)  sq.  ft. 

14.  84pt.  =  (?)qt.  19.  72in.  =  (?)ft. 

15.  84qt.  =  (?)pt.  20.  72in.  =  (?)yd. 

21.  Make  out  an  imaginary  personal  account  of  six  items 
on  each  side,  and  balance  the  account. 

22.  Write  a  bill  for  silver  purchased;  a  cash  check  for 
merchandise ;   a  receipted  bill  for  furniture  bought ;    an. 
invoice  of  a  wholesale  dealer. 


76  ARITHMETIC  OF  INDUSTRY 

Exercise  57.    Problems  without  Numbers 

1.  If  you  know  the  wages  per  hour  and  the  number  of 
hours  worked  each  day  by  each  man,  without  overtime, 
how  do  you  find  the  total  wages  due   all  the  men  in  a 
shop  in  a  week? 

2.  If  you  know  the  regular  wages  due  a  man  per  hour, 
the  wages  for  overtime,  and  the  number  of  hours  he  works 
each  day  in  a  week,  some  of  these  being  overtime,  how  do 
you  find  his  total  wages  for  a  week  ? 

3.  If  you  know  the  weight  of  a  steel  girder  per  running 
foot  and  the  length  of  the  girder,  how  do  you  find  the 
total  weight? 

4.  If  you  know  the  weight  of  a  girder  and  its  length, 
how  do  you  find  its  weight  per  running  foot  ? 

5.  If  you  know  the  weight  of  each  of  several  bales  of 
cotton  and  the  price  paid  per  pound,   how  do  you  find 
the  gain  or  loss  to  a  firm  that  buys  this  cotton  on  a  basis 
of  500  Ib.  to  the  bale  ? 

6.  If  you  know  the  shipping  rate  per  hundredweight 
from  where  you  live  to  Liverpool,  and  know  the  weight  of 
a  shipment,  how  do  you  find  the  total  charge  for  freight  ? 

7.  If  a  farmer  has  wheat  to  ship  to  Chicago  and  knows 
the  freight  rate  per  bushel  or  carload  and  the  number  of 
bushels  or  carloads,  how  does  he  find  the  freight  charges  ? 

8.  If  a  farmer  knows  the  charges  for  shipping  a  certain 
number  of  bushels  of  corn  to  Chicago,  how  does  he  find 
the  freight  rate  per  bushel? 

9.  If  you  know  the  average  weight  of  a  fleece  of  wool 
and  the  number  of  fleeces  a  sheep  grower  has,  how  do 
you  find  the  number  of  bales  of  wool  weighing  250  Ib. 
«ach  and  the  amount,  if  any,  that  will  be  left  over  ? 


ARITHMETIC  OF  THE  BANK  77 

V.    ARITHMETIC  OF  THE  BANK 

Saving.  A  boy  who  puts  1$  each  day  into  a  toy  bank 
will  have  enough  in  six  months  to  buy  a  catcher's  mitt  or 
three  baseball  bats.  A  girl  who  saves  100  a  day  will  have 
enough  in  a  month  or  two  to  buy  a  good  pair  of  shoes. 
A  man  who  saves  $1  a  day  will  have  enough  in  a  few 
years  to  buy  a  building  lot  in  some  place  where  he  may 
care  to  live.  We  often  get  the  things  that  we  need  for 
comfortable  living  or  the  things  that  give  us  legitimate 
pleasure,  by  saving  a  little  at  a  time. 

The  following  suggestion  to  teachers  will  be  found  helpful : 
Begin  with  a  brief  discussion  of  the  need  and  value  of  saving 
money.  There  are  many  of  us  who  never  learn  how  to  save  money 
wisely.  Many  of  us  prefer  to  gratify  our  immediate  desires  rather 
than  to  provide  for  the  future..  Why  is  this  a  bad  plan?  On  the 
other  hand,  there  are  others  of  us  who,  in  order  to  save  for  the  future, 
deny  ourselves  the  things  which  it  would  be  real  economy  to  buy. 
There  is  always  the  temptation  to  live  extravagantly.  Extravagance 
includes  not  only  living  beyond  our  means  but  also  spending  money 
foolishly.  Every  boy  and  every  girl  should  begin  early  in  life  to  form 
the  habit  of  saving,  no  matter  if  it  be  but  a  few  dollars  a  year. 

i 

Exercise  58.    Saving 

1  A  man  wishes  to  buy  an  automobile  that  costs  $780. 
If  he  saves  |2  every  week  day,  in  how  many  weeks  will 
he  save  enough  to  buy  the  car  ? 

2.  A  boy  wishes  to  buy  a  camera  that  costs  |6.60.    If 
he  saves  10$  every  week  day,  in  how  many  wieeks  will  he 
save  enough  to  buy  the  camera  ? 

3.  A  girl  wishes  to  buy  a  purse  that  costs  60$.    If  she 
saves  5$  every  week  day,  in  how  many  weeks  will  she  save 
enough  to  buy  the  purse  ? 


78  ARITHMETIC  OF  THE  BANK 

4.  A  man  who  has  been  smoking  six   cigars  a  day, 
which  he  buys  at  the  rate  of  three  for  a  quarter,  decides 
to  give  up  smoking  and  save  the  money.    How  much  will 
this  saving  amount  to  in  5  yr.  ? 

In  all  such  problems  the  year  is  to  be  considered  as  365  da.  (or 
313  da.,  excluding  Sundays),  although  in  5  yr.  there  will  probably  be 
one  leap  year  and  there  may  be  two  leap  years,  and  although  a  year 
need  not  have  exactly  52  Sundays. 

5.  A  boy  earns  some  money  by  selling  papers.   He  finds 
that  he  can  easily  save  150  a  day,  excluding  Sundays.    If 
he  does  this  for  5  yr.,  how  much  will  he  save  in  all  ? 

6.  A  woman  in  a  city  has  a  telephone  for  which  she  is 
charged  5$  for  each  call.    She  finds  that  she  can  economize 
by  reducing   the    number   of   her  telephone   calls   on  an 
average  four  a  day,  including  Sundays.    If  she  does  this 
for  3  yr.,  how  much  will  she  save  ? 

Make  out  accounts,  inserting  dates,  items  of  receipts  and 
payments,  and  the  balances,  given  the  following: 

7.  On  hand,  $4.20.    Receipts:   200,  400,  250,  $1,  300, 
650.    Payments:   250,  300,  100,  750. 

8.  On   hand,    $4.30.    Receipts:    100,   150,  '600,    550. 
Payments:  300,  420,  750,  50,  300,  600,  200,  50. 

9.  On  hand,  $4.63.  Receipts:  100, 120,  320, 120,  $1.10. 
Payments:   250,  350,  50,  240,  $1.50,  250,  500. 

10.  On  hand,  $5.10.     Receipts:   $1.25,  350,  750,  420, 
680,  700,   $1.22.    Payments:   $1.50,  700,  50,  100. 

11.  On  hand,  $1.30.     Receipts:  $1,  400,  50,  150,  100, 
250,  120,  160,  100.   Payments:  100,  200,  50,  50,  220,  30. 

12.  On  hand,  $3.20.    Receipts:  700,  600,  100,  30,  150, 
100,  250,  50,  50,  300,  520,  300.    Payments:  $1,50,  500, 
900,  700,  180,  120,  300,  450,  250. 


SAVINGS  BANKS  79 

Bank  Account  Essential.  One  thing  that  is  essential  at 
some  time  to  everyone  who  hopes  to  succeed  is  a  bank 
account.  A  reliable  person  may  "open  an  account,"  as  it 
is  called,  as  soon  as  he  begins  to  save  even  small  amounts. 

People  who  are  saving  money  usually  keep  it  in  a  bank 
until  they  have  enough  for  investing  permanently.  Certain 
kinds  of  banks,  such  as  savings  banks  and  trust  companies, 
not  only  guarantee  to  take  care  of  all  money  left  with  them 
by  depositors  but  also  pay  a  certain  per  cent  of  interest. 
National  banks  also  generally  allow  interest  on  what  are 
called  inactive  accounts  ;  that  is,  deposits  that  remain  undis- 
turbed for  some  time. 

Many  schools  have  found  it  interesting  and  profitable  to  organize 
school  banks,  electing  the  officers  and  carrying  on  a  regular  banking 
business,  either  with  small  amounts  of  real  money  placed  on  deposit 
by  students  and  transferred  by  the  teacher  to  some  bank  or  trust 
company,  or  with  imitation  money.  Such  exercises  should  not,  how- 
ever, interfere  with  the  work  in  computing. 

Savings  Bank.  To  deposit  money  a  person  goes  to  a 
»ank,  says  that  he  wishes  to  open  an  account,  and  leaves 
his  money  with  the  officer  in  charge.  The  officer  gives 
him  a  book  in  which  is  written  the  amount  deposited, 
and  the  depositor  writes  his  name  in  a  book  or  on  a  card, 
for  identification.  When  he  wishes  to  draw  out  money, 
he  takes  his  book  to  the  bank,  signs  a  receipt  or  a  check 
for  the  amount  he  desires,  and  receives  the  money,  the 
amount  being  entered  in  his  book. 

Students  should  be  told  of  the  advantages  of  opening  even  small 
accounts  at  a  savings  bank.  A  boy  who  deposits  $1  a  week  for  10  yr. 
in  a  bank  paying  2%  every  6  mo.  and  adding  it  to  the  account,  will 
have  $631.54  in  10  yr.,  and  a  man  who  puts  in  $10  a  week  will  have 
about  ten  times  as  much,  or  $6317.16. 

The  class  should  be  told  about  trust  companies,  which  take  charge 
of  funds,  manage  estates,  and  pay  interest  on  deposits. 


80  ARITHMETIC  OF  THE  BANK 

Exercise  59.    Saving 

1.  How  much  will  25$  saved  each  working  day,  310  such 
days  to  the  year,  amount  to  in  10  yr.? 

2.  If  a  boy,  beginning  at  the  age  of  14  yr.,  saves  25$ 
a  day  for  310  da.  a  year  and  deposits  it  in  a  bank,  how 
much  has  he  when  he  is  21  yr.  old,  not  counting  interest  ? 

Interest  and  withdrawals  are  not  to  be  considered  in  such  cases. 

3.  If  a  man  saves  $3.25  a  week  out  of  his  wages  and 
continues  to  do  this  52  wk.  in  a  year  for  12  yr.,  how  much 
money  will  he  save  ? 

4.  If  a  father  gives  his  daughter  on  each  birthday  until 
and  including  the  day  she  is  25  yr.  old  as  many  dollars  as 
she  is  years  old,  depositing  it  for  her  in  a  savings  bank, 
how  much  has  she  when  she  is  25  yr.  old  ? 

5.  A  merchant  saves  $750  the  first  year  he  is  in  business. 
The  second  year  he  saves  one  third  more  than  in  the  first 
year.    The  third  year  his  savings  are  only  85%  as  much  as 
the  second  year.    The  fourth  year  they  increase  30  %  over 
the  third  year.    How  much  does  he  save  in  the  four  years? 

6.  A  man  works  on  a  salary  of  $18  a  week  for  52  wk. 
in  a  year.    His  expenses  are  $3.75  a  week  for  house  rent, 
60%  as  much  for  clothing,  300%  as  much  for  food  as  for 
clothing,  and  20  %  as  much  for  other  necessary  expenses  as 
for  food.   How  much  of  his  salary  can  he  deposit  each  year 
in  the  savings  bank  ? 

7.  A  clerk  had  a  salary  of  $12  a  week  two  years  ago  and 
a  commission  of  2  %  on  goods  he  sold.  That  year  he  worked 
50  wk.  and  sold  $4800  worth  of  goods.    Last  year  his  salary 
was  increased  25%,  his  rate  of  commission  remaining  the 
same.    He  worked  48  wk.  and  sold  goods  to  the  amount 
of  $5000.    How  much  was  his  income  increased  ? 


INTEREST  81 

Interest.  If  Mr.  James  has  a  house  and  lot  worth 
|5000,  and  rents  it  to  Mr.  Jacobs  at  $42  a  month,  his 
income  from  the  rent  is  12  x  $42,  or  $504  a  year.  This 
is  a  little  more  than  10%  of  the  value  of  the  property, 
but  out  of  it  Mr.  James  has  to  pay  for  various  expenses, 
such  as  insurance,  repairs,  and  taxes. 

If  Mr.  James  has  $5000  and  lends  it  to  Mr.  Jacobs  at 
the  rate  of  6%  a  year,  his  income  from  this  transaction  is 
6%  of  $5000,  or  $300  a  year. 

Money  paid  for  the  use  of  money  is  called  interest.  In  the 
above  illustration  about  the  lending  of  money  $300  is  the 
interest  for  1  yr.,  6%  is  the  rate  of  interest,  and  $5000  is 
the  principal. 

Schools  do  not  require,  as  formerly,  the  learning  of  many  defini- 
tions. What  is  necessary  is  that  the  student  should  use  intelligently 
such  terms  as  interest,  rate,  and  principal. 

Men  often  have  to  borrow  money  to  carry  on  their  busi- 
ness. For  example,  a  merchant  may  wish  to  buy  a  lot  of 
holiday  goods,  feeling  sure  that  he  can  sell  them  at  a 
profit.  In  this  case  it  is  good  business  for  him  to  borrow 
the  money,  say  in  November,  taking  advantage  of  all  cash 
discounts  allowed,  and  then  to  repay  the  money  in  January 
after  the  goods  are  sold.  If,  for  example,  he  needs  $1000 
for  2  mo.  and  can  borrow  it  from  a  bank  at  the  rate  of 
6%  a  year,  he  will  have  to  pay  ^  of  6%  of  $1000,  or  1% 
of  $1000,  or  $10,  a  sum  which  he  can  e'asily  afford  to  pay 
for  the  use  of  the  money. 

To  find  the  interest  on  any  sum  of  money  for  part  of  a 
year  first  find  the  interest  for  1  yr.  and  then  find  it  for  the 
given  part  of  a  year. 

Formal  rules  for  such  work  need  not  be  memorized.  An  example 
^r  two  may  profitably  be  worked  on  the  blackboard  before  studying 
the  next  page. 


82  ARITHMETIC  OF  THE  BANK 

Exercise  60.    Interest 

Examples  1  to  27,  oral 

Find  the  interest  on  the  following  amounts  for  1  yr.  at  the 
given  rates : 

1.  $1000,  5%.  6.  $150,  4%.  ll.  $400,  6%. 

2.  $1000,  6%.  7.  $250,  4%.  12.  $400, 

3.  $1000,4^%.  8.  $500,  5%.  13.  $1000, 

4.  $2000,  5%.  9.  $600,  6%.  14.  $1000, 

5.  $3000,  6%.  10.  $800,  3£%.  15.  $1000,  4f  %. 

.Frnrf  iAe  interest  on  the  folloiving  amounts  for  6  mo.  at 
the  given  rates : 

16.  $100,  6%.    18.  $1000,  5%.   20.  $3000,  4%. 

17.  $300,  6%.     19.  $2000,  5%.   21. 


e  interest  .on  the  following  amounts  for  1  yr.  6  mo. 
at  the  given  rates : 

22.  $1000,  4%.        24.  $2000,  6%.        26.  $5000,  4%. 

23.  $1000,  5%.        25.  $5000,  5%.       27.  $2500,  4%. 

28.  A   man  having    $17,250   invested   in  business  has 
found  that  his  net  profits   average  16%   a  year  on  the 
investment.    He  is  offered  $25,000  for  the  business,  and 
he  could  invest  the  money  at  4^-%.    If  he  sells  out  and 
retires,  what  is  his  annual  loss  in  income  ? 

29.  In  April  a  coal  dealer  borrowed   $66,420  at  5%. 
With  this  he  purchased  his  summer's   supply  of  coal  at 
$5.40  a  ton,  his  overhead  charges  being  30  <£  a  ton.    He 
sold  the  coal  at  $6.68  a  ton,  the  buyers  paying  for  the 
unloading  and  delivery,  and  he  paid  his  debt  in  October 
after  keeping  the  money  6  mo.    How  much  did  he  gain  ? 


INTEREST  FOE  MONTHS  AND  DAYS  88 

Interest  for  Months  and  Days.  Suppose  that  a  man  bor- 
rows from  a  bank  $400  on  Sept.  10,  1919,  at  6%.  What 
will  the  interest  amount  to  Aug.  7,  1920  ? 

yr.  mo.        da. 

1920         8         7  =  second  date 
1919          9       10  =  first  date 

10       27  =  difference  in  time 

Taking,  as  is  usual,  30  da.  to  the  month,  the  difference 
in  time  is  327  da.  We  therefore  have 

109 


Hence  the  interest  due  August  7,  1920,  is  $21.80. 

Banks  usually  lend  money  for  a  definite  number  of  days  or  else 
require  payment  to  be  made  on  demand.  In  either  case  they  com- 
pute the  interest  for  the  days  that  the  borrower  has  the  money  and 
not  for  months  and  days.  To  enable  them  to  compute  the  interest 
easily  they  have  interest  tables.  Private  individuals,  however,  occa- 
sionally have  to  compute  interest  for  months  and  days,  and  in  that 
case  they  may  proceed  as  in  the  above  problem. 

It  is  a  waste  of  time  for  the  student  to  find  the  interest  on  very 
small  or  very  large  sums  of  money,  for  very  short  or  very  long 
periods,  or  at  more  than  legal  rates.  A  few  such  examples  may  be 
given,  however,  for  practice  in  computation.  In  general,  interest  is 
now  reckoned  on  such  a  sum  as  $750  rather  than  $749.75,  and  for 
periods  not  exceeding  90  da.  rather  than  one  involving  years,  months, 
and  days.  Teachers  should  advise  the  students  that  if  the  interest 
is  for  more  than  1  yr.  they  should  first  find  it  for  the  given  number 
of  years,  and  then,  by  the  above  method,  for  parts  of  a  year.  Such 
cases  are,  however,  rapidly  becoming  obsolete.  Banking  facilities 
make  it  rare  to  find  interest  periods  for  years,  months,  a»d  days. 


84  ARITHMETIC  OF  THE  BANK 

Interest  for  30,  60,  and  90  Days.  In  borrowing  money 
at  a  bank  the  time  for  which  the  money  is  borrowed  is 
usually  30  da.,  60  da.,  or  90  da.,  except  when  repayment 
is  to  be  made  on  demand.  Since  6  %  is  the  most  common 
rate,  it  is  convenient  to  be  able  to  work  mentally  the 
common  types  of  interest  examples. 

How  much  interest  must  you  pay  if  you  borrow  $500 
from  a  bank  at  6%  for  60  da.?  for  30  da.?  for  90  da.? 

Since  60  da.  =  -£$$  yr.  =  ^  yr.,  the  interest  on  $500  for  60  da.  is 
£  of  6%  of  $500,  or  1%  of  $5( 


For  30  da.  the  interest  is  \  of  1%  of  $500,  or  $2.50. 
For  90  da.  the  interest  is  f  of  1%  of  $500,  or  $7.50. 

From  this  work  state  a  simple  rule  for  finding  the  interest 
at  6%  for  60  da.?  for  30  da.?  for  90  da.? 

Exercise  61.    Interest 

Examples  1  to  15,  oral 
Find  the  interest  at  6%  on  the  following  amounts: 

1.  |400,  for  60  da.  6.  $840,  for  60  da. 

2.  $650,  for  30  da.  7.  $350,  for  90  da. 

3.  $725,  for  60  da.  8.  $450,  for  30  da. 

4.  $875,  for  60  da.  9.  $950,  for  30  da. 

5.  $900,  for  90  da.  10.  $860,  for  30  da. 

11.  Find  the  interest  on  $600  for  60  da.  at  5%. 
The  interest  at  6%  is  $6,  and  so  at  5%  it  is  $  of  $6. 

12.  Find  the  interest  on  $3000  for  30  da.  at  5%. 

13.  Find  the  interest  on  $240  for  90  da.  at  5%. 

14.  Find  the  interest  on  $600  for  60  da.  at  4%. 

15.  Find  the  interest  on  $1200  for  30  da.  at  3%. 


INTEREST  85 

16.  Find  the  interest  on  |400  for  2  yr.  10  mo.  27  da. 
at  6%. 

The  interest  for  2  yr.  is  2  x  6%  of  $400,  or  $48,  and  for  10  mo.  27  da. 
is  $21.80,  as  found  on  page  83.  Hence  the  total  interest  is  $69.80. 

As  already  stated,  such  examples  are  becoming  more  rare.  A  few 
are  given  on  this  page,  chiefly  as  exercises  in  computation. 

Find  the  interest  on  the  following : 

17.  $1250  for  2  mo.  17  da.  at  5%. 

18.  |1500  for  7  mo.  23  da.  at  6%  ;  at  5J%. 

19.  $2400  for  8  mo.  11  da.  at  5%  ;  at  6%  ;  at  5J$. 

20.  $575  for  2  yr.  9  mo.  15  da.  at  5%  ;  at  5J%. 

21.  $850  for  3  yr.  10  mo.  6  da.  at  5J%  ;  at  6%. 

22.  $925  for  4  yr.  10  mo.  6  da.  at  6%  ;  at  5%. 

23.  A  dealer  bought  24  sets  of  furniture  on  Nov.  1,  at 
$50  a  set,  promising  to  pay  for  them  later,  with  interest  at 
6%.     He  paid  the  bill  on  the  following  Jan.  16.     What 
was  the  amount  of  principal  and  interest  ? 

24.  A  man  borrowed  $750  on  Mar.  10,  at  6%,  and  $1600 
on  Apr.  10,  at  5%.    He  paid  the  entire  debt  on  July  10  of 
the  same  year.    How  much  did  he  pay  in  all  ? 

25.  A  man  borrowed  $750  on  May  1,  at  5%,  and  $1800 
on  July  5,  at  4^%.    He  paid  both  debts  with  interest  on^  ( 
Dec.  16  of  the  same  year.    How  much  did  he  pay  in  all  ? 

26.  What  is  the  total  amount  of  principal  and  interest 
on  $950  borrowed  Mar.  10,  at  6%,  and  $1600  borrowed 
May  15,  at  5%,  the  payment  in  both  cases  being  made 
on  Oct.  20  of  the  same  year? 

27.  A  man  borrowed  $750  on  May  9,  at  5%,  and  $625 
on  June  15,  at  6%,  each  loan  to  run  for  60  da.     When 
was  each  due,  and  how  much  was  the  total  interest? 


86 


ARITHMETIC  OF  THE  BANK 


$2000.=  first  principal 
.02 

).=  int.  first  6  mo. 


Interest  at  Savings  Banks.  Savings  banks  usually  pay 
interest  every  six  months  or  every  three  months.  This 
interest  is  added  to  the  principal,  and  the  total  amount 
then  draws  interest. 

Compound  Interest.  When  interest  as  it  becomes  due  is 
added  to  the  principal  and  the  total  amount  then  draws 
interest,  the  investor  is  said  to  receive  compound  interest 
on  his  money. 

Compound  interest 
is  not  commonly  used, 
but  if  one  collects  in- 
terest when  due  and  at 
once  reinvests  it,  he 
practically  has  the  ad- 
vantage of  compound 
interest.  The  method 
of  finding  compound 
interest  is  substantially 
the  same  as  that  used 
in  simple  interest. 

For  example,  how 
much  is  the  amount  of 
$2000  in  2  yr.,  deposi- 
ted in  a  savings  bank 
that  pays  4%  annu- 
ally, the  interest  being 
compounded  semiannu- 
ally?  How  much  is  the 
compound  interest  ? 


2000. 
$2040.=  amt.  after  6  mo. 

.02 

$40.80  =  int.  second  6  mo. 
2040. 
$2080.80  =  amt.  after  1  yr. 

.02 

$41.62  =  int.  third  6  mo. 
2080.80 
$2122.42  =  amt.  after  11  yr. 

.02. 

$42.45  =  int.  fourth  6  mo. 
2122.42 

$2164.87  =  amt.  after  2  yr. 
2000. 
$164.87  =  int.  for  2  yr. 


Simple  interest  for  the 
same  time  is  $160,  or  $4.87  less  than  the  compound  interest. 

Here  the  compound  interest  has  been  found  exactly,  but  savings 
banks  pay  interest  only  on  the  dollars  and  not  on  the  cents. 


8T 


Savings  Bank  Account.  The  following  is  a  specimen 
account  at  a  savings  bank  which  pays  interest  at  the  rate 
of  4  °f0  a  year, .  the  interest  being  payable  semiannually, 
on  January  1  and  July  1,  on  the  smallest  balance  on 
deposit  at  any  time  during  the  previous  interest  period: 


DATE 

DEPOSITS 

INTEREST 

PAYMENTS 

BALANCE 

1922 

July 

1 

600 

50 

600 

50 

July 

20 

75 

675 

50 

Sept. 

6 

120 

555 

50 

Dec. 

7 

60 

615 

50 

Dec. 

20 

65 

550 

50 

1918 

Jan. 

1 

11 

561 

50 

May 

9 

200 

761 

50 

July 

1 

11 

22 

772 

72 

The  smallest  balance  during  the  first  interest  period  is 
1550.50.  Interest  is  computed  on  the  dollars  only,  the 
cents  being  neglected.  At  4%  per  year  the  interest  for 
6  mo.  on  $550  is  2%  of  $550,  or  $11.  In  the  second  period 
the  smallest  balance  is  $561.50,  and  therefore  the  interest 
is  2%  of  $561,  or  $11.22. 

Some  banks  allow  interest  from  the  first  of  each  month; 
others  from  the  first  of  each  quarter ;  others,  as  above, 
from  the  first  of  each  half  year.  The  interest  is  computed 
on  the  smallest  balance  on  hand  between  this  day  and  the 
next  interest  day,  and  is  usually  added  every  half  year, 
although  it  is  sometimes  added  every  quarter. 

Students  should  ascertain  the  local  custom  as  to  savings  banks. 


88  AEITHMETIC  OF  THE  BANK 

Exercise  62.   Compound  Interest 

Find  the  amount  of  principal  and  interest  at  simple  interest, 
and  also  at  interest  compounded  in  a  savings  bank  annually  : 

1.  13000,  2yr.,  5%.  6.  $2750,  4  yr., 

2.  $3000,  4yr.,  6%.  7.  $825.50,  5yr., 

3.  $2000,  4yr.,  4%.  8.  $2000,  6  yr.,  4%. 

4.  $3250,  4yr.,  3%.  9.  $625.50,  4  yr., 

5.  $3750,  4yr.,  3%.  10.  $875.50,  3  yr., 


the  amount  of  principal  and  interest,  the  interest  beiny 
compounded  in  a  savings  bank  st-miannually  : 

11.  $400,  3yr.,  4%.  16.  $600,  2  yr.,  4%. 

12.  $600,  2yr.,  4%.jte,ffU1*17.  $2000,  2  yr.,  4J%. 

13.  $850,  2yr.,  6%.  £(p$2000,  3  yr.,  4%. 

14.  $900,  3yr.,  3%.  ,  19.  $3000,  2  yr.,  3£%. 

15.  $900,  3yr.,  4%.*Sbl     'tfeo.  $3000,  4  yr.,  4%. 

21.  If  you  deposited  $140  in  a  savings  bank  on  July  17, 
1919,  and  $35  on  Feb.  9,  1920,  and  if  you  have  made  no 
withdrawals,  to  how  much  interest  are  you  entitled  July  1, 
1920?    In  this  bank  on  July  1  and  Jan.  1  interest  on  each 
deposit  at  4%  per  year  is  credited  from  the  day  of  deposit 
if  on  the  first  day  of  a  month,  and  otherwise  from   the 
first  day  of  the  following  month. 

22.  If  a  man  deposits  $1500  in  a  savings  bank  on  Jan.  1, 
$215  on  Feb.  1,  $140  on  May  7,  $270  on  Sept.  11,  and  $243 
on  Dec.  3,  and  makes  no  withdrawals,  how  much  will  he 
have  to  his  credit  on  the  following  Jan.  1  ?   In  this  bank  on 
July  1  and  Jan.  1  interest  on  each  deposit  at  4%  per  year  is 
credited  from  the  day  of  deposit  if  on  the  first  day  of  a  month, 
and  otherwise  from  the  first  day  of  the  following  month. 


POSTAL  SAVINGS  BANKS  89 

Postal  Savings  Bank.  The  United  States  government 
conducts  a  savings  bank  in  connection  with  the  post  office. 
Although  all  savings  banks  are  carefully  regulated  and  in- 
spected by  the  state  governments,  there  are  many  persons 
who  are  willing  to  take  the  smaller  rate  of  income  which 
the  postal  savings  bank  pays,  because  of  the  fact  that  our 
government  guarantees  the  payment  of  their  money. 

Any  person  of  the  age  of  10  yr.  or  over  may  deposit 
money  in  amounts  of  not  less  than  $1,  but  no  fractions 
of  a  dollar  are  accepted  for  deposit.  No  one  can  deposit 
more  than  $1000  in  any  one  calendar  month  or  have  a 
balance  at  any  time  of  more  than  $1000,  exclusive  of 
accumulated  interest.  Deposits  may  be  made  at  the  larger 
post  offices,  and  a  depositor  receives  a  postal  savings  certifi- 
cate for  the  amount  of  each  deposit.  Interest  is  paid  by  the 
government  at  the  rate  of  2%  for  each  full  year  that  the 
money  remains  on  deposit,  beginning  on  the  first  day  of 
the  month  next  following  the  one  in  which  the  deposit  is 
made.  Interest  is  not  paid  for  any  fraction  of  a  year. 
A  person  may  exchange  his  deposits  in  sums  of  $20  or 
multiples  of  $20  for  bonds  bearing  interest  at 


Exercise  63.   Postal  Savings  Bank 

All  work  oral 

Find  the,  interest  for  1  yr.  on  the  following  deposits  : 
1.  $30.        2.  $40.         3.  $75.         4.  $300.       5.  $500. 

Find  the  interest  for  2  yr.  on  the  following  deposits: 
6.  $50.         7.  $60.         8.  $100.       9.  $200.     10.  $500. 

Find  the  interest  for  1  yr.  on  a  %\°]o  bond  of: 

11.  $80.   12.  $240.  13.  $360.  14.  $480.  15.  $500. 


90 


ARITHMETIC  OF  THE  BANK 


Bank  of  Deposit.  When  a  man  has  money  enough  ahead 
to  pay  liis  bills  by  checks,  he  will  find  it  convenient  to 
have  an  account  with  a  bank  such  as  merchants  commonly 
use,  sometimes  called  a  bank  of  deposit. 

Such  banks  do  not  pay  interest  on  small  accounts,  the 
deposit  being  a  matter  of  convenience  and  safety.  If  a 
man  wishes  to  open  an  account  he  sometimes  has  to  give 
references,  for  banks  do  not  wish  to  do  business  with 
unreliable  persons.  A  man's  credit  in  business  is  always 
a  valuable  asset. 

In  some  sections  of  the  country  banks  receive  deposits 
under  two  classes  of  accounts,  savings  accounts  and  check- 
ing accounts.  In  the  former  case  they  act  as  savings 
banks ;  in  the  latter,  as  banks  of  deposit.  For  the  purposes 
of  the  school  it 
is  not  necessary 
to  consider  this 
difference  further. 
Students  should, 
however,  investi- 
gate the  local  cus- 
tom in  the  matter. 

Deposit  Slip.  A 

man,  when  he  de- 
posits money  or 
checks  in  a  bank, 
fills  out  a  deposit 
slip  similar  to  the 
one  here  shown. 

Sometimes  the  depositor  enters  the  name  of  the  bank  on  which 
each  check  is  drawn ;  sometimes  the  receiving  teller  at  the  bank  does 
this  by  writing  the  bank's  number ;  and  sometimes  it  is  not  entered 
at  all.  These  are  technicalities  that  do  not  concern  the  school. 


DEPOSITED  FOR  CREDIT  OF 

IN  THE 

SECOND   NATIONAL  BANK 

OF  THE  CITY  OF  NEW  YORK 
IQ1 

RTTT.S 

DOLLARS 

CENTS 

roiN 

CT-fFCTC  ON                             R'K 

BANKS  OF  DEPOSIT  91 

Exercise  64.   Deposit  Slips 

Write  or  fill  out  deposit  slips  for  the  following  deposits, 
inserting  the  name  of  the  depositor  and  of  the  bank  : 

1.  Bills,  $375;  silver,  $60;   check  on  Garfield  Bank, 
$87.50;  check  on  Miners  Bank,  $627.75. 

2.  Bills,  $423;  gold,  $175;  silver,  $235.75;  check  on 
Corn  Exchange  Bank,  $736.90. 

3.  Bills,   $135;    check   on    Second   National   Bank  of 
New  York,  $425 ;  check  on  Chase  National  Bank,  $75.40. 

4.  Bills,  $1726;  gold,   $100;   silver,   $200;   check  on 
Merchants  Bank,  $245.50;  check  on  Union  Bank,  $275.40. 

5.  Bills,  $1275 ;  checks  on  Harriman  National  Bank, 
$146.50,  $200 ;  checks  on  Jefferson  Bank,  $325,  $86.50. 

6.  Gold,  $100 ;  checks  on  First  National  Bank,  $175, 
$240,  $32.80 ;  checks  on  Sherman  Bank,  $37.42,  $61.85. 

7.  Bills,  $2475 ;  silver,  $275.50 ;  check  on  Case  Bank, 
$43.50 ;  check  on  Miners  National  Bank,  $250. 

8.  Bills,  $345;  silver,  $350.75;  gold,  $480;  check  on 
Merchants  National  Bank,  $455 ;  check  on  Farmers  Trust 
Co.,  $262.50 ;  check  on  City  Bank,  $1000. 

9.  A  man  deposited  $475.75  in  cash  to-day,  a  check  for 
50%  of  a  debt  of  $675  due  him,  and  a  check  in  payment 
for  45yd.  of  velvet  at  $2.25  a  yard  less  33|-%  discount. 
Make  out  a  deposit  slip. 

10.  A  merchant  received  cash  for  8  doz.  forks  @  $14.75, 
5-|-  doz.  teaspoons  @  $13,  a  watch  costing  $40.50,  and 
4  clocks  @  $7.75.  He  also  received  a  check  on  the  Lincoln 
Trust  Co.  for  3  doz.  dessert  spoons  @  $17.75  and  4-|  doz. 
nutcrackers  @  $9.  He  deposited  all  this  in  a  bank.  Make 
out  a  deposit  slip. 


JMl 


92 


ARITHMETIC  OF  THE  BANK 


Check.  A  check  book  containing  checks  and  stubs,  substan- 
tially as  follows,  although  often  varying  in  certain  details, 
is  given  the  depositor  when  he  opens  an  account. 


No.  8<?6 


New  York, 

Jtatumal 


to  the  order  of.. 


^Dollars 


fC. 


NO. 


CHECK 

The  person  to  whom  a  check  is 
payable  is  called  the  payee.  In  the 
above  example  Myron  P.  Jones  is 
the  payee.  A  check  may  be  made 
payable  to  "  Self,"  in  which  case 
the  drawer  alone  can  collect  it ;  or 
to  the  order  of  the  payee,  as  in  the 
above  check,  in  which  case  the  payee 
must  indorse  it,  that  is,  he  must 
write  his  name  across  the  back ;  or 
to  the  payee  or  "  bearer,"  or  to 

'"  Cash,"  in  which  cases  anyone  can  collect  it. 

The  indorsement  made  by  Mr.  Jones  would  appear  on  the 

back  in  the  form  here  shown : 


To 


For 


Amt. 


STUB 


The  teacher  should  explain  to  the 
class  the  nature  of  checks,  the  different 
ways  of  filling  them  out,  and  the  end 
on  which  they  should  be  indorsed.  The  teacher  should  explain  the 
advantages  of  the  various  methods  of  making  the  checks  payable 
and  the  students  should  write  or  fill  out  various  styles  of  checks. 


CHECKS  93 

Exercise  65.    Bank  Deposits 

1.  If  your  deposits  in  a  bank  have  been  $58.65,  $43, 
$25,  $80,  $95,  $25.75,  $12.50,  and  $9.50,  and  you  have 
drawn  checks  for  $8.25,  $16.30,'  $15.75,  $16.48,  and  $25, 
what  is  then  your  balance  at  the  bank  ? 

2.  A  man  earning  $22.50  a  week  deposits  $15  every 
Saturday,  and  each  Monday  gives  a  check  for  $4.50'  for 
his  board.    What  will  be  his  balance  in  13  wk.? 

3.  If  your  deposits  in  a  bank  have  been  $68.45,  $92.30, 
$47.60,  $38.50,  $78.75,  and  $96.70,  and  you  have  drawn 
checks  for  $8.55,  $23.65,  $8.58,  $48.75,  and  $34.60,  what 
is  your  balance  ? 

4.  A  merchant   having   $980.75   in   the   bank   deposits 
during  the  next  week  $185.50,  $97.85,  $135.50,  $86.85, 
and  $236.80.    He  gives  checks  for  $89.65,  $37.20,  $93.60, 
$15.20,  $248.70,  and  $39.80.    What  is  now  his  balance? 

5.  A  man  having   $825.60  in  the  bank  gives  a  check 
for  $128.75.   He  then  deposits  checks  for  $75.80,  $126.75, 
$234.80,  and  $42.80.     During  this  time  he  gives  checks 
for  $125.80  and  $24.75.    What  is  now  his  balance? 

6.  A   merchant   having    $828.50   in   the  bank  deposits 
$567.80,  $245.50,  $89.65,  $482.86,  $429.50,  and  $376.50, 
and  draws  checks  for  $427.50,  $38.95,  $67.82,  $568.70,  and 
$122.58.    He  also  pays  by  check  a  bill  for  $125.40  less 
10%,  another  bill  for  $86  less  4%,  and  another  for  $48.75 
less  6%.    What  is  now  his  balance? 

7.  A  merchant  having  $1026.92  in  the  bank  deposits 

$488.75,  $928.75,  $386.48,  $442.80,  $196.85,  $327.75, 
and  draws  checks  for  $96.75,.  $286.75,  ,$342.80,  $438.50. 
He  also  gives  a  check  for  $230  plus  interest  for  4  mo.  at 
5°/0.  What  is  now  his  balance  ? 


94  AEITHMETIC  OF  THE  BANK 

Promissory  Note.  A  paper  signed  by  a  borrower,  agree-, 
ing  to  repay  a  specified  sum  of  money  on  demand  or  at  a 
specified  time,  is  called  a  promissory  note,  or  simply  a  note. 

The  sum  borrowed  is  called  the  principal,  or,  if  a  note 
is  given,  the  face  of  the  note. 

The  sum  of  the  principal  (or  face)  and  the  interest  is 
called  the  amount  of  the  note. 

A  note  should  state  the  date,  face,  rate,  person  to  whom 
payable,  and  time  to  run  (time  before  it  is  due  to  be  paid), 
and  that  it  has  been  given  for  value  received  by  the  maker. 
The  following  is  a  common  form  for  a  time  note : 


$75.—  NEW  YORK,  &&(wua,vy  7, 

c/u^  ttuw£^*  after  date,  for  value  received,  I  promise 

to  pay  to jlo-fm  jtoAnA&n or  order, 

0  -j-  ft  r  00  T~\       1 1 

&s/v-£/ytvu~ir-i^-£'  • —  Dollars 

with  interest  at  5%. 


The  following  is  a  common  form  for  a  demand  note : 


$50.-                                  NEW  YORK,  l?la,y  2, 
On  demand,  for  value  received,  I  promise  to  pay  to 
/?.  jSwea. or  order, 


with  interest  at 


PEOMISSOEY  NOTES  95 

Parties  to  a  Note.  The  person  named  in  a  note  as  the 
one  to  whom  it  is  payable  is  called  the  payee.  The  person 
who  signs  a  note  is  called  the  maker. 

Indorsing  a  Note.  If  the  payee  sells  the  note,  he  must, 
when  it  is  payable  to  himself  or  order,  indorse  it. 

A  note  is  indorsed  by  the  payee  by  writing  his  name 
across  the  back.  The  indorser  must  pay  the  note  if  the 
maker  does  not. 

A  note  payable  to  John  Johnson  or  bearer  may  be  sold 
without  indorsement.  Such  notes  are  not  common. 

If  the  payee  wishes  to  sell  the  note  without  being  respon- 
sible for  the  payment  in  case  the  maker  should  fail  to  pay 
it,  he  may  write  the  words  "without  recourse"  across  the 
back,  and  write  his  name  underneath.  This  means  that  he 
relinquishes  all  title  to  it  and  that  the  buyer  cannot  come 
back  (have  recourse)  on  him.  The  following  are  the  forms : 

INDORSEMENT  IN  BLANK  INDORSEMENT  IN  FULL    LIMITED  INDORSEMENT 


jlo/iru 


Teachers  should  explain  fully  the  meaning  of  these  several  indorse- 
ments, and  should  have  the  students  indorse  notes  properly. 

Rate  of  Interest.  The  United  States  borrows  money  at 
rates  of  about  3%  to  3-|%.  Savings  banks  pay  depositors 
about  3%  or  4%.  In  cities,  on  good  security,  borrowers 
usually  pay  from  4%  to  6%. 

When  a  note  bears  interest,  but  the  rate  is  not  specified, 
it  bears  interest  at  a  certain  rate  fixed  by  the  law  of  the 
state.  In  many  states  this  rate  is  6  °/0. 

In  most  states,  if  a  note  falls  due  on  a  Sunday  or  a  legal 
holiday,  it  is  payable  on  the  next  business  day. 


96  ARITHMETIC  OF  THE  BANK 

Exercise  66.   Promissory  Notes 

1.  Compute  the  amount  of  the  first  note  on  page  94. 

2.  Write  a  promissory  note,  signed  by  A  and  payable 
to  B,  for  $75,  due  in  1  yr.,  at  6%.    Find  the  amount. 

3.  F.  H.  Ryder  borrows  $750,  at  6%,  for  1  yr.,  from 
M.  P.  Read.    He  gives  a  note  payable  to  Mr.  Read  or 
order.    Mr.  Read  sells  the  note  to  F.  N.  Cole.    Make  out 
the  note,  indorse  it  in  full,  and  find  the  amount. 

4.  Make  out  a  note  like  the  one  referred  to  in  Ex.  3, 
but  for  $725.     Indorse  it  in  blank  and  find  the  amount 
due  at  the  end  of  the  year.   Write  a  check  for  this  amount. 

5.  Make  out  a  note  like  the  one  referred  to  in  Ex.  3, 
but  for  $1250.    Indorse  it  without  recourse  and  find  the 
amount  due  at  the  end  of  the  year.    Write  a  check  for 
this  amount. 

6.  Write  a  note  for  $275,  bearing  interest  at  6%  and 
payable  in  6  mo.    Insert  names  and  dates,  and  indorse  it 
payable  to  the  order  of  John  Ball,  with  a  second  indorse- 
ment by  which  Mr.  Ball  transfers  it  to  James  Clay. 

Make  out  and  indorse,  payable  to  the  order  of  the  buyer, 
the  following  notes,  and  find  the  amount  due  on  each : 


7. 
8. 
9. 
10. 
11. 
12. 
13. 

MAKER 

A.  N.  Cole 
A.  R.  Doe 
D.E.Bell 
O.N.Olds 
S.  M.  Roe 
A.  J.  Burr 
A.  B.  Bain 

PAYEE 

P.  R.  Carr 
E.  F.  Dun 
E.  L.  Cree 
B.R.Hall 
C.  N.  King 
G.F.Ray 
G.  F.  Dow 

J. 
M 
A. 
B. 
O. 
A 
F. 

BUYER 

R.Hall 
.  L.  King 
K.  lies 
S.Hill 
.M.Coe 
.  R.  Carr 
E.Trae 

FACE 

$775 
$650 
$550 
$350 
$225 
$850 
$950 

RATE 

6% 
5% 
6% 
5% 
6% 

4« 

TIME 
4  mo. 
8  mo. 
3yr. 
5  mo. 
2yr. 
6  mo. 
9  mo. 

BANK  DISCOUNT  97 

Bank  Discount.  When  a  man  borrows  from  a  bank  on  a 
time  note  he  pays  the  interest  in  advance.  Interest  is  not 
mentioned  in  the  note,  because  it  has  already  been  paid. 

Interest  paid  in  advance  on  a  note  is  called  discount. 

Teachers  should  call  the  attention  of  the  students  to  the  fact  that 
the  same  word  is  used  for  bank  and  commercial  (trade)  discount, 
explaining  that  the  mathematical  process  is  the  same  in  both  cases ; 
that  is,  finding  some  per  cent  of  a  number. 

Unless  otherwise  directed,  always  call  30  da.  a  month. 

Proceeds.  The  face  of  a  note  less  the  discount  is  called 
the  proceeds. 

What  are  the  discount  and  proceeds  of  a  note  for  $225 
for  6  mo.  at  5%  ? 

The  discount  (interest)  for  1  yr.  is  5%  of  $225,  or  $11.25. 
The  discount  for  6  mo.  is  £  of  $11.25,  or  $5.63. 
The  proceeds  are  $225  -  $5.63,  or  $219.37. 

Exercise  67.  Bank  Discount 

Find  the  discounts  and  the  proceeds  on  the  following : 

1.  1300,  1  mo.,  Q%.  11.  $300,  60  da.,  5%. 

2.  $500,  30  da.,  6%.  12.  $575,  2  mo., 

3.  $750,  2  mo.,  5%.  13.  $400,  4  mo., 

4.  $475,  3  mo.,  6%.  14.  $800,  2  mo.,  3£%. 

5.  $825,  2  mo.,  5%.  15.  $5000,  63  da.,  5%. 

6.  $500,  90  da.,  6%.  16.  -$3350,  93  da.,  6%. 

7.  $475,  6  mo.,  6%.  17.  $1250,  10  da.,  Q%. 

8.  $800,  90  da.,  6%.  18.  $2500,  15  da.,  6%. 

9.  $150,  45  da.,  6%.  19.  $1500,  20  da.,  6%. 
10.  $600,  1  mo.,  5J%.  20.  $1250,  45  da.,  5%. 


98  ARITHMETIC  OF  THE  BANK 

Find  the  discounts  and  the  proceeds  on  the  following : 

21.  $675,  30  da.,  Q%.  27.  $3000,  90  da.,  5%. 

22.  $750,  90  da.,  5%.  28.  $4500,  90  da., 

23.  $850,  30  da.,  6%.  29.  $3750,  30  da., 

24.  $3500,  60  da.,  5%.  30.  $136.75,  30  da.,  6%. 

25.  $4250,  60  da.,  Q%.  31.  $275.50,  60  da.,  5%. 

26.  $4500,  90  da.,  5%.  32.  $42,000,  30  da.,  5%. 

33.  Make  out  a  60-day  note  for  $450,  dated  to-day,  pay- 
able to  R.  D.  Cole's  order  at  some  bank.   Discount  it  at  6%. 

34.  Make  out  a  30-day  note  for  $350,  dated  to-day,  pay- 
able to  Frank  Lee's  order  at  some  bank.    Discount  it  at  5%. 

35.  Make  out  a  60-day  note  for  $960,  dated  to-day,  pay- 
able to  Ray  Lang's  order  at  some^bank.    Discount  it  at  5%. 

36.  Make  out  a  90-day  note  for  $3000,  dated  to-day, 
payable  to  L.  D.  Baldwin's  order  at  some  bank  of  which 
you  know.    Discount  it  at  6%. 

37.  A  man's  bank  account  shows  deposits  of  $175.50, 
$68.50,  $50,  $300,  $40,  $75,  $100,  $125,  and  $500 ;  checks 
drawn,  $43.75,  $125.50,  $62,  $5,  and  $125.35.    He  needs 
$4500  to  start  him  in  business  and  wishes  to  keep  about 
$500  in  the  bank.    How  much  money,  to  the  nearest  $100, 
should  he  borrow  ? 

38.  If  the  man  in  Ex.  37  makes  out  a  note  for  this  amount 
for  90  da.  at  6%,  how  much  discount  must  he  pay  ?   What 
are  the  proceeds  ?  What  are  the  proceeds  for  60  da.? 

39.  A.  D.  Redmond  has  to  pay  a  debt  of  $2000  less  10%. 
He  has  in  the  bank  $587.60,  and  has  $327.50  in  cash  in 
his  safe.    He  wishes  to  leave  about  $500  in  the  bank  and 
about  $100  in  his  safe.    How  much,  to  the  nearest 
must  he  borrow?    Discount  the  note  for  30  da.  at  6 


DISCOUNTING  NOTES  99 

Commercial  Paper.  If  a  dealer  buys  some  goods  for  the 
fall  trade,  but  does  not  wish  to  pay  for  them  until  after 
the  holidays,  he  may  buy  them  on  credit,  giving  his  note. 
The  manufacturer  may  need  the  money  at  once,  in  which 
case  he  will  indorse  the  note  and  sell  it  to  a  bank  or  to  a 
note  broker  for  the  face  less  the  discount.  Such  notes  are 
commonly  called  commercial  paper. 

For  example,  if  you  give  a  manufacturer  your  note  for 
$500,  dated  Sept.  1  and  due  Jan.  1,  with  interest  at  5%, 
and  he,  needing  the  money,  discounts  the  note  at  a  bank 
Sept.  1  at  6%,  what  are  the  proceeds? 

Face  of  the  note $500. 

Interest  for  4  mo.  at  5% 8.33 

Amount  due  at  maturity  .....  $508.33 

Discount  for  4  mo.  at  6%      .     .     .     .  10.17 

Proceeds $498.16 

The  manufacturer  may  not  need  the  money  Sept.  1, 
and  so  he  may  put  the  note  away, in  his  safe  and  let  it 
lie  there  drawing  interest.  But  if  he  needs  the  money 
Sept.  16  he  may  then  decide  to  discount  the  note  at  a 
bank.  We  shall  then  have 

Face  of  the  note      .......  $500. 

Interest  for  4  mo.  at  5% 8.33 

Amount  due  at  maturity $508.33 

Discount  for  10 7  da.  at  6%  .     .     .     .  9.07 

Proceeds $499.26 

Banks  usually  compute  the  discount  period  in  days, 
and  the  discount  by  tables  based  on  360  da.  to  the  year. 

If  the  banks  themselves  need  more  money,  they  may  rediscount 
this  paper  at  the  Federal  Reserve  Bank.  The  details  of  the  Federal 
Reserve  Bank  need  not  be  considered  in  the  schools. 


100  ARITHMETIC  OF  THE  BANK 

Six  Per  Cent  Method.  The  following  short  method,  com- 
monly known  as  the  Six  Per  Cent  Method,  has  been  referred 
to  already  (page  84),  and  is  convenient  not  only  in  com- 
puting interest  but  also  in  discounting  notes. 

Find  the  interest  on  |420  for  5  mo.  10  da.  at  6%. 

Since  2  mo.=  ^  yr.,  the  rate  for  2  mo.  is  -^  of  6%,  or  1%. 

The  interest  for  2  mo.  is  1%  of  $420         =    $4.20 
The  interest  for  2  mo.  more  =      4.20 

The  interest  for  1  mo.  more  is  -|  of  $4.20  =      2.10 
The  interest  for  10  da.  is  J  of  $2.10  m  .70 

The  interest  for  5  mo.  10  da.  =  $11.20 

Therefore  the  interest  at  6%  for  60  days  is  0.01  of  the 
principal,  for  6  days  is  0.001  of  the  principal,  and  for  other 
periods  the  interest  can  be  found  from  this  interest. 

The  rule  is  conveniently  stated  as  follows: 

For  30  da.  take  ±  of  1%  ;  for  60  da.,  1%  ;  for  90  da.,  Ij  %. 

Since  bank  notes  usually  run  for  30  da.,  60  da.,  or  90  da.,  since 
6%  is  the  most  common  rate,  and  since  we  can  tell  the  discount  for 
60  da.  by  simply  glancing  at  the  face  of  the  note,  we  can  often  find 
mentally  the  discount  on  bank  notes  for  the  usual  periods. 

Exercise  68.   Six  Per  Cent  Method 

1.  Find  the  interest  on  $4250  for  60  da.  at  6%,  first  by 
the  Six  Per  Cent  Method,  then  by  cancellation,  and  finally 
by  the  ordinary  method  of  finding  the  interest  for  1  yr.  and 
then  for  the  fractional  part  of  a  year.    Write  a  statement 
tailing  the  advantage  of  the  Six  Per  Cent  Method. 

2.  Using  the  three  methods  mentioned  in  Ex.  1,  find 
the  interest  at  6%  on  $875  for  90  da.;  on  $2500  for  30  da. 

3.  A  note  for  $1275  is  discounted  for  60  da.  at  6%.   Find 
the  discount  and  the  proceeds. 


SIX  PEll  CENT  METHOD  101 

Find  the  discounts  at6°/0  on  notes  for  the  following  amounts: 

4.  $3000,  for  90  da.  10.  $250,  for  3  mo.  8  da. 

5.  $2550,  for  90  da.  11.  $800,  for  3  mo.  15  da. 

6.  $4575,  for  30  da.  12.  $750,  for  1  mo.  18  da. 

7.  $3575,  for  90  da.  13.  $950,  for  3  mo.  20  da. 

8.  $4625,  for  90  da.  14.  $2175,  for  3  mo.  15  da. 

9.  $8250,  for  30  da;  15.  $6500,  for  1  mo.  18  da. 

16.  A  note  for  $1500  is  discounted  for  30  da.  at  6%. 
Find  the  discount  and  the  proceeds. 

17.  A  note  for  $3750  is  discounted  for  90  da.  at  6%. 
Find  the  discount  and  the  proceeds. 

18.  A  note  for  $1250  is  discounted  for  90  da.  at  5%. 
Find  the  discount  and  the  proceeds. 

Find  the  discount  at  6%  and  deduct  ^  of  this. 

19.  A   man  wishes  to  borrow  about  $7500  for  60  da. 
The  bank  offers  to  lend  it  to  him  at  5%.    If  he  makes  out 
a  note  for  $7600  and  discounts  this,  how  much  more  than 
$7500  will  he  receive  from  the  bank  ? 

20.  A  speculator  buys  some  property  for  $30,000.    He 
pays  $9600  down  and  borrows  the  balance  for  90  da,  at  5%. 
How  much  discount  must  he  pay  on  the  note  ? 

21.  How  much  greater,  if  any,  is  the  discount  on  a  note 
for  $2500  discounted  for  60  da.  at  6%  than  on  one  for 
$5000  discounted  for  30  da.  at  6%? 

22.  A  man  needs  $9750  to  pay  for  some  goods.    If  he 
gives  a  note  for  $9800  for  30  da.  at  6%,  will  the  proceeds 
be  more  or  will  they  be  less  than  the  amount  he  needs  ? 
How  much  more  or  how  much  less? 

23.  A  firm  gives  its  note  for  $12,500,  discounting  it  for 
90  da.  at  5-|%.    How  much  is  the  discount? 


102 


ARITHMETIC  OF  THE  BANK 


Exercise  69.   Miscellaneous  Problems 

1.  Make   out  a   60-day  note  for    $950,   dated   to-day, 
payable  to  M.  W.  Gross  or  order  at  some  bank  in  your 
vicinity,  sign  it  X.  Y.  Z.,  and  discount  it  at  6%. 

2.  Fill  out  the  blanks  in  a  table  like  the  following  and 
compute  the  discount  at  6%  on  all  the  notes  mentioned: 


TIME  IN 
DAYS 

TO   BE 
FOUND 

FACE  OF  NOTE 

$90 

$280 

$575 

$850 

$1200 

30 

Discount 
Proceeds 

60 

Discount 
Proceeds 

90 

Discount 
Proceeds 

3.  George  Lang  sold  his  farm  of  120  A.  to  Fred  Ray  at 
$95  an  acre.    Ray  paid  $8000  in  cash  and  gave  a  90-day 
note  without  interest  for  the  balance.    If  Lang  discounted 
the  note  at  6%  the  day  it  was  made,  how  much  did  Lang 
actually  receive  for  the  farm  ? 

4.  A  man  deposits  $420  in  a  savings  bank  on  July  1, 
$48.50  on  July  19,  $41.30  on  Aug.  9,  and  $72.90  on  Dec.  7. 
ffis  withdrawals  are  $20.50  on  July  29,  and  $51  on  Dec.  22. 
The  next  year  he  deposits  $39.80  on  Feb.  4  and  $126.40 
on  Apr.  14,  withdrawing  $38.50  on  Feb.  23.    The  savings 
bank  pays  1%  every  three  months,  on  Jan.  1,  Apr.  1,  July  1, 
and  Oct.  1,  on  the  smallest  balance  in  even  dollars  during 
the  preceding  quarter.    Find  the  man's  balance  on  Oct.  1 
following  his  last  deposit. 


EEVIEW  103 

Exercise  70.   Review  Drill 

Write  in  common  numerals : 
1.  Seven  thirty-seconds.         2.  Ninety-six  thousandths. 

Add,  and  also  subtract : 

3.                             4.                               5.  6. 

4346.8                9185.48  4878.46  4008.06 

3946.8                 7369.72  2398.59  869.58 

Multiply,  and  also  divide : 
7.  259.2  by  2.88.   8.  946.96  by  6.23.   9.  95.19  by  5.01. 

Find  the  sum,  difference,  product,  and  both  quotients  of: 
10.  f,  -f.      11.   f,  f.      12.  |,  f.      13.   ^  1       14.   |,  1|. 
By  both  quotients  of  |  and  3  is  meant  f  ^-  §  and  §  -^-  f  • 

Find  the  interest  on  the  folloiving : 

15.  $275,  2yr.,  6%.  16.   $275,  2  yr.  8  mo.,  6%. 

Find  the  discounts  on  the  folloiving  bills : 

17.  |475,  6%.  18.  $8734.75,  6%,  3%. 

19.  Find  the  discount  on  $725  for  60  da.  at  4%. 

20.  Find  the  interest  on  $12,500  for  5  mo.  at  4-|%. 

21.  Find  the  interest  on  $675  for  1  yr.  8  da.  at  Q%. 

22.  A  workman  has  $1250  in  the  savings  bank  Jan.  1, 
on  which  he  receives  3-|%  interest.    At  the  end  of  6  mo. 
he  takes  out  this  money  and  puts  the  $1250  with  accumu- 
lated interest  in  another  bank  where  he  receives  4  %  interest. 
How  much  has  he  to  his  credit  after  the  money  has  been 
in  the  second  bank  for  6  mo.  ? 


104  ARITHMETIC  OF  THE  BANK 

Exercise  71.   Problems  without  Numbers 

1.  Given  the  face,  rate,  and  time,  how  do  you  ascertain 
the  interest  due  on  a  note  ? 

2.  Which  pays  the  better  interest,  if  the  money  is  left 
undisturbed  for  a  given  number  of  years,  a  promissory  note 
or  a  savings-bank  deposit  at  the  same  rate  per  year?    Why  ? 

3.  How  do  you  fill  out  a  deposit  slip  ?    After  entering 
the  items,  what  operation  do  you  perform  ?    How  do  you 
make  sure  that  the  result  is  correct? 

4.  If  you  know  a  man's  balance  in  a  bank  a  week  ago 
and  his  deposits  and  checks  since,  how  do  you  find  his 
balance  now? 

5.  How  do  you  find  the  discount  on  a  promissory  note  ? 
How  do  you  find  the  proceeds  ? 

6.  If  a  note  drawing  a  certain  rate  of  interest  is  dis- 
counted on  the  day  it  is  made,  at  the  same  rate,  are  the 
proceeds  greater  than  the  face,  or  equal  to  it,  or  less  ? 
Why  is  this  ? 

7.  How  can  a  manufacturer  discount  a  claim  against 
a  purchaser,  the  claim  not  being  yet  due  ?     How  is  the 
discount  found  ? 

8.  If  you  know  the  proceeds  and  the  discount,  how  do 
you  find  the  face  of  a  note  ? 

9.  If  you  know  the  face  of  a  note  and  the  proceeds, 
how  do  you  find  the  discount? 

10.  If  you  know  the  face  of  a  note,  the  proceeds,  and 
the  time,  how  do  you  find  the  rate  of  discount'? 

11.  If  you  have  money  in  a  postal  savings  bank,  how 
much  higher  rate  of  interest  will  you  receive  if  you  ex- 
change it  for  government  bonds  ? 


EXERCISE   1 

Taking  (a)  8.32,  (b)  124.8,  (c)  16.64,  (d)  0.208, 
(e)  2.496,  or  (f)  0.2912,  as  the  teacher  directs: 

1.  Add  it  to  0.732  +  9  +  7.29  +  68.4+1.726  +  0.85. 

2.  Subtract  it  from  25.865  +  21.854  +  78.146. 

3.  Multiply  it  by  125,  using  a  short  method. 

4.  Divide  it  by  0.13. 

5.  Find  £  of  it;  87J%  of  it. 

EXERCISE   2 

Taking  (a)  $8.64,  (b)  $12.96,  (c)  £Z7.0S,  (d)  $£/.00, 
(e)  $25.92,  or  (f)  $38.88,  as  the  teacher  directs: 

1.  Add  it  to  $15  +  $0.76  +  $2.88  +  $9.36  +  $2.75. 

2.  Subtract  it  from  $2.63  +  $8.13  +  $20.75  +  $16.87. 

3.  Multiply  it  by  66|>  using  a  short  method. 

4.  Divide  it  by  $1.08. 

5.  Find  12J%  of  it;  37£%  of  it;  62|%  of  it 

This  Material  for  Daily  Drill  is  so  arranged  as  to  give  daily 
practice  in  the  fundamental  operations.  By  first  going  through  all 
the  exercises  with  the  number  denoted  by  (a),  and  then  with  the 
one  denoted  by  (b),  and  so  on,  more  than  a  hundred  different 
exercises  will  result,  or  more  than  one  exercise  for  each  school  day 
of  the  half  year,  giving  enough  for  a  selection. 

105 


106  MATERIAL  FOE  DAILY  DKILL 

EXERCISE  3 

Taking  (a)  $8.96,  (b)  $13.44,  (c)  $17.92,  (d)  $22.40, 
(e)  $26.88,  or  (f)  $31.36,  as  the  teacher  directs  : 

1.  Add  it  to  |19  +  1287.30  +  $2.75  +  148.60  -f  142.86. 

2.  Subtract  it  from  $4.63  +  $10.14  +  $27.82  +  $9.86. 

3.  Multiply  it  by  750,  using  a  short  method. 

4.  Divide  it  by  $1.12. 

5.  Find  25%  of  it;  2J%  of  it;  250%  of  it. 

EXERCISE  4 

Taking  (a)  $9.28,  (b)  $13.92,  (c)  $18.56,  (d)  $23.20, 

(e)  $27.84,  or  (f)  $32.48,  as  the  teacher  directs : 

1.  Add  it  to  $37.62  +  $0.27  +  $150  +  $3.98  +  $48.60. 

2.  Subtract  it  from  $25.37  +  $17.26  +  $14.96  +  $5.04. 

3.  Multiply  it  by  125,  using  a  short  method. 

4.  Divide  it  by  $1.16. 

5.  Find  |  of  it;  75%  of  it;  f  of  it;  37^%  of  it;  3.75%  of  it. 

EXERCISE  5 

Taking  (a)  $9.92,  (b)  $14.88,  (c)  $19.84,  (d)  $24.80, 

(e)  $29.76,  or  (f)  $34.72,  as  the  teacher  directs : 

1.  Add  it  to  $3.09  +  $17+  $0.75  +  $27.68  +  $9.32. 

2.  Subtract  it  from  $2.80  +  $15.06  +  $19.87+  $10.13. 

3.  Multiply  it  by  37-|,  using  a  short  method. 

4.  Divide  it  by  $1.24. 

5.  Divide  it  by  f ;  by  2|. 


MATERIAL  FOR  DAILY  DRILL  10T 

EXERCISE  6 

Taking  (a)  $10.56,  (b)  $15.84,  (c)  $21.12,  (d)  $26.40, 
(e)  $31.68,  or  (f)  $36.96,  as  the  teacher  directs  : 

1.  Add  it  to  $0.29  +  $28.70  +  $15  +  $3.28  +  $4.96. 

2.  Subtract  it  from  $140  +  $72.36  +  $27.64. 

3.  Multiply  it  by  33-^,  using  a  short  method. 

4.  Divide  it  by  8  ;  by  33  ;  by  $2.64. 

5.  Divide  it  by 


EXERCISE  7 

Taking  (a)  $1038,  (b)  $1632,  (c)  $21.76,  (d)  $27.20, 
(e)  $32.64,  or  (f  )  $38.08,  as  the  teacher  directs  : 

1.  Add  it  to  $7.33  +  $26  +  $0.48  +  $7.88  +  $2.94. 

2.  Subtract  it  from  $75  +  $37.42  +  $12.58. 

3.  Multiply  it  by  6.25. 

4.  Divide  it  by  8  ;  by  17;  by  34;  by  $2.72;  by  $1.36. 

5.  Divide  it  by  6  j  ;  by  ll£. 

EXERCISE  8 

Taking  (a)  $11.20,  (b)  $16.80,  (c)  $22.40,  (d)  $33.60, 
(e)  $28,  or  (f)  $39.20,  as  the  teacher  directs  : 

1.  Add  it  to  9  times  itself,  using  a  short  method. 

2.  Subtract  it  from  11  times  itself. 

3.  Multiply  it  by  37^. 

4.  Divide  it  by  8  ;  by  7  ;  by  5  ;  by  35  ;  by  $1.40. 
.     5.  Divide  it  by  8f;  by  4|;  by 


108  MATERIAL  FOR  DAILY  DRILL 

EXERCISE  9 

Taking  (a)  $11.52,  (b)  $17.28,  (c)  $23.04,  (d)  $28.80, 
(e)  $34.56,  or  (f  )  $40.32,  as  the  teacher  directs  : 

1.  Add  it  to  $1.20  +  $0.92  +  $17  +  $3.75  +  $28.67. 

2.  Subtract  it  from  $32.75  +  $19.82  +  $10'.18. 

3.  Multiply  it  by  62.5. 

4.  Divide  it  by  2  ;  by  4  ;  by  8  ;  by  16  ;  by  32  ;  by  36. 

5.  Multiply  it  by  -|.    Divide  it  by  2|^. 

EXERCISE  10 

Taking  (a)  $11.84,  (b)  $77.76,  (c)  $23.68,  (d)  $29.60, 

(e)  $35.52,  or  (f)  $41.44,  as  the  teacher  directs  : 

1.  Add  it  to  $12  +  $16.75  +  $0.82  +  $2.98  -f  $48.20. 

2.  Subtract  it  from  $75  +  $37.80  +  $42.60  +  $17.90. 

3.  Multiply  it  by  37.85. 

4.  Divide  it  by  2;  by  4  ;  by  8  ;  by  $0.37. 

5.  Multiply  it  by  0.12J;  by  J;  by 


EXERCISE   11 

Taking  (a)   1216,  (b)  182.4,   (c)   24.32,   (d)    0.4256, 
(e)  3.648,  or  (f)  0.304,  as  the  teacher  directs  : 

1.  Add  it  to  9  +  15.75  +  21  +  5J. 

2.  Subtract  it  from  1300. 

3.  Multiply  it  by  0.365. 

4.  Divide  it  by  3.04  ;  by  0.7  ;  by  40  ;  by  400  ;  by  4000. 

5.  Divide  it  by  12  J;  by  6J;  by  3£. 


MATERIAL  FOR  DAILY  DRILL  109 

EXERCISE    12 

Taking    (a)    124.8,    (b)   18.72,   (c)    24.96,    (&)  0.312, 
(e)  3.744,  or  (f)  0.4368,  as  the  teacher  directs: 

1.  Add  it 'to  3.848  +  148.276+175  +  48.76  +  9.009. 

2.  Subtract  it  from  4.6273+74.896  +  56.215. 

3.  Multiply  it  by  12.5,  using  a  short  method. 

4.  Divide  it  by  2  ;  by  4  ;  by  8  ;  by  13  ;  by  3.12  ;  by  6£. 

5.  Find  12-|%  of  it,  using  a- short  method. 

EXERCISE  13 

Taking  (a)  13.44,   (b)  20.16,  (c)  2.688,   (d)  0.4704, 
(e)  4.032,  or  (f)  0336,  as  the  teacher  directs: 

1.  Add  it  to  72.8796+182.08  +  7.087+72.6  +  0.983. 

2.  Subtract  it  from  2.786  +  46.93  +  53.17. 

3.  Multiply  it  by  342.87. 

4.  Divide  it  by  3  ;  by  7 ;  by  8 ;  by  0.042 ;  by  3.36. 

5.  Of  what  number  is  it  75%  ?  f  ?  7J%?  f%  ? 

EXERCISE   14 

Zfc&ingr    (a)  1.408,  (b)  07..Z0,   (c)    £.&?£,    (d)    0.352, 
(e)  4.004,   or  (f)  0.4928,  as  the  teacher  directs: 

1.  Add  it  to  482.76894  +  9  +  0.987  +  0.7236  +  483. 

2.  Subtract  it  from  0.7+276.93  +  14.963  +  5.037. 

3.  Multiply  it  by  f  of  J. 

4.  Divide  it  by  0.8;  by  4.4;  by  0.352;  by  2.2;  by  f. 

5.  Of  what  number  is  it  80%  ?  120%  ?  1J  ?  f  ? 


110  MATERIAL  FOR  DAILY  DRILL 

EXERCISE    15 

Taking    (a)  15.36,    (b)   23.04,    (c)  3.072,    (d)  0.384, 
(e)  4.608,  or  (f)  0.5376,  as  the  teacher  directs: 

1.  Add  it  to  0.2702  +  298.742  +  0.7298  +  7017+  2983. 

2.  Subtract  it  from  36.7+921.006+78.239  +  21.761. 

3.  Multiply  it  by  122J. 

4.  Divide  it  by  9|. 

5.  Of  what  number  is  it  125%?  1J?  £?  12j%? 

EXERCISE   16 

Taking    (a)  15.68,    (b)  2.352,    (c)   313.6,    (d)   0.392, 
(e)  470.4,  or  (f)  5488,  as  the  teacher  directs: 

1.  Add  it  to  0.1271+  2789.762  +  2936  +  7064  +  0.8729. 

2.  Subtract  it  from  48.789  +  968.32  +  3429  +  6571. 

3.  Multiply  it  by  itself. 

4.  Divide  it  by  0.0784. 

5.  What  per  cent  is  it  of  5  times  itself  ?   of  half  itself  ? 

EXERCISE    17 

Taking    (a)    2.88,    (b)    26.4,     (c)    124.8,    (d)    17.04, 
(e)  34.56,  or  (f)  69.12,  as  the  teacher  directs: 

1.  Add  it  to  125%  of  itself. 

2.  Subtract  it  from  200%  of  itself. 

3.  Multiply  it  by  0.5%  of  itself. 

4.  Divide  it  by  0.15 ;  by  0.075 ;  by  1.5. 

5.  Of  what  number  is  it  12%?  1.2%?  120%? 


PART  II.   GEOMETRY 
I.    GEOMETRY  OF  FORM 

First  Steps  in  Geometry.  Thousands  of  years  ago,  when 
people  began  to  study  about  forms,  they  were  interested  in 
pictures  showing  the  shapes  of  objects ;  these  they  used  in 
decorating  their  walls,  and  later  in  showing  the  plans  of 
their  houses  and  their  temples  and  in  representing  animals 
and  human  beings.  As  land  became  valuable  they  showed 
an  interest  in  measuring  objects,  fields,  and  building  ma- 
terial. When  they  wished  to  locate  places  on  the  earth's 
surface  and  when  they  began  to  study  the  stars,  it  was 
necessary  that  they  should  consider  position.  From  very 
early  times,  therefore,  the  ideas  of  form,  size,  and  position 
have  interested  humanity. 

There  are  three  things  which  we  naturally  ask  about 
an  object :  What  is  its  shape  ?  How  large  is  it  ?  Where 
is  it  ?  It  is  these  three  questions  that  form  the  bases  of 
the  kind  of  geometry  which  we  are  now  about  to  study. 
There  are  also  other  questions  which  we  might  ask  about 
the  object,  such  as  these :  How  much  is  it  worth  ?  What 
is  its  color  ?  Of  what  is  it  made  ?  None  of  these  questions, 
however,  has  to  do  with  geometry. 

The  teacher  will  recognize  that  demonstrative  geometry  is  not 
touched  upon  directly  by  the  three  questions  above  set  forth. 
Another  question  might  be  asked  relating  to  all  three,  namely, 
How  do  you  know  that  your  statement  is  true?  It  is  this  question 
which  leads  to  the  proof  of  propositions.  For  the  present  we  are 
concerned  almost  exclusively  with  intuitional  and  observational 
geometry  as  related  to  the  questions  of  shape,  size,  and  position. 

Ill 


112  GEOMETRY  OF  FORM 

Geometric  Figures.  You  are  already  familiar  with  such 
common  forms  as  the  square,  triangle,  circle,  arc,  and  cube. 
Such  forms  are  generally  known  as  geometric  figures. 

Angle.  Two  straight  lines  drawn  from  a  point  form  an 
angle.  The  two  straight  lines  are  called  the  sides  of  the 
angle,  and  the  point  where  they  meet  is  called  the  vertex. 

The  three  most  important  angles  are  the  right  angle, 
the  acute  angle,  which  is  less  than  a  right  angle,  and  the 
obtuse  angle,  which  is  greater  than  a  right  angle. 


RIGHT  ANGLE  ACUTE  ANGLE  OBTUSE  ANGLE 

If  necessary,  the  teacher  should  explain  what  we  mean  when 
we  say  that  an  angle  is  greater  than  or  less  than  another  angle. 
This  is  easily  done  by  slowly  opening  a  pair  of  compasses. 

Acute  angles  and  obtuse  angles  are  called  oblique  angles. 

Triangle.  A  figure  bounded  by  three  straight  lines  is 
called  a  triangle. 


EQUILATERAL     ISOSCELES  RIGHT  ACUTE  OBTUSE 


The  five  most  important  kinds  of  triangles  are  the  equi- 
lateral triangle,  having  all  three  sides  equal;  the  isosceles 
triangle,  having  two  sides  equal ;  the  right  triangle,  having 
one  right  angle ;  the  acute  triangle,  having  three  acute 
angles;  and  the  obtuse  triangle,  having  one  obtuse  angle. 

The  side  opposite  the  right  angle  in  a  right  triangle  is 
called  the  hypotenuse  of  the  right  triangle. 

The  sum  of  the  sides  of  a  triangle  is  called  the  perimeter. 


ANGLES  AND  TRIANGLES  113 

Exercise  1.   Angles  and  Triangles 

Examples  1  to  6,  oral 

1.  Point  to  three  right  angles  in  the  room. 

2.  Point,  if  possible,  to  two  straight  lines  on  the  wall 
or  on  a  desk  which  form  an  acute  angle. 

3.  Point,  if  possible,  to  two  straight  lines  hi  the  school- 
room which  form  an  obtuse  angle. 

4.  Which  is  the  greater,  an  acute  angle  or  an  obtuse 
angle  ? 

5.  How  many  right  angles  all  lying  flat  on  the  top  of 
a  table  will  completely  fill  the  space  around  a  point  on 
the  table? 

6.  If  one  side  of  an  equilateral  triangle  is  6  in.,  what 
.is  the  perimeter  of  the  triangle  ? 

7.  Draw  a  right  angle  as  accurately  as  you  can  by  the 
aid  of  a  ruler. 

8.  Draw  an  acute  angle  and  an  obtuse  angle,  writing 
the  name  under  each. 

9.  In  this  figure  name 
by  capital  letters  the  tri- 
angles which  seem  to  you 
to  be  right  triangles. 

10.  In  the  same  figure  name  by  a  small  letter  each  of 
the  acute  angles. 

11.  In  the  same  figure  name  by  a  capital  letter  each  of 
the  obtuse  triangles. 

12.  What  kinds  of  angles  are  represented  hi  the  figure 
by  the  letters  o,  p,  r,  s  ?    Write  the  name  after  each  letter. 

13.  If  two  straight  lines  intersect,  what  can  you  say  as 
to  any  equal  angles  ? 


114  GEOMETRY  OF  FORM 

Quadrilateral.  A  figure  bounded  by  four  straight  lines 
is  called  a  quadrilateral. 

The  rectangle,  square,  parallelogram,  and  trapezoid,  the 
four  most  important  kinds  of  quadrilaterals,  are  shown  below. 


RECTANGLE      SQUARE    PARALLELOGRAM    TRAPEZOID 

A  quadrilateral  which  has  all  its  angles  right  angles  is 
called  a  rectangle. 

A  rectangle  which  has  its  sides  all  equal  is  called  a 
square. 

A  quadrilateral  which  has  its  opposite  sides  parallel  is 
called  a  parallelogram. 

A  quadrilateral  which  has  one  pair  of  opposite  sides 
parallel  is  called  a  trapezoid. 

It  is  not  necessary  at  this  time  to  give  a  formal  definition  of 
parallel  lines.  The  students  are  familiar  with  the  term. 

We  shall  hereafter  use  the  word  line  to  mean  straight  line  unless 
we  wish  to  use  the  word  straight  with  line  for  purposes  of  emphasis. 

Polygon.  A  figure  bounded  by  straight  lines  is  called  a 
polygon.  The  quadrilaterals  shown  above  are  all  special 
kinds  of  polygons,  and  a  triangle  is  also  a  polygon. 

Polygons  may  have  three,  four,  five,  six,  or  any  other 
number  of  sides  greater  than  two. 

The  side  on  which  a  polygon  appears  to  rest  is  called 
the  base  of  the  polygon. 

The  sum  of  all  the  sides  of  a  polygon  is  called  the 
perimeter  of  the  polygon. 

The  points  in  which  each  pair  of  adjacent  sides  intersect 
are  called  the  vertices  of  the  polygon. 

In  the  case  of  a  triangle  the  vertex  of  the  angle  opposite 
the  base  is  usually  called  the  vertex  of  the  triangle. 


115 


Congruent  Figures.   If  two  figures  have  exactly  the  same 
shape  and  size,  they  are  called  conc/ruent  figures. 

Drawing  Instruments.    The  instruments  commonly  used 
in    drawing  the  figures  in  geometry  are   the   compasses, 
the  ruler,  the  protractor,  and  the  right 
triangle.     The    compasses    are    used 
for     drawing     circles     as     here 
shown  and  also  for  laying 
distances  on  paper. 
A   protractor    of   the    general    type 
here  .shown    is    convenient  for 
use  by  students,  and  with 
its  aid  angles  ot  any 
number     of     de- 
grees can  be 
drawn. 


0^2 


\\ 


i\ 


\\ 


For  work  out  of  doors  a  surveyor  measures  angles  and 
finds  levels  by  means  of  a  transit  such  as  is  here  shown. 

Each  student  should  have  a  ruler, 
a  pair  of  compasses,  and  a  protractor, 
since  the  constructions  studied  in  this 
book  can  be  made  only  by  their  use. 

If  necessary  such  familiar  terms 
as  circle,  radius,  diameter,  arc,  and 
circumference  should  be  explained 
informally.  They  are  more  for- 
mally stated  later. 

On  pages  117  and  119  and  later 
in  the  work  some  interesting  illus- 
trations of  ancient  instruments  are 
given.  Students  often  make  similar 
instruments  for  use  in  geometry. 


116  GEOMETRY  OF  FORM 

Constructing  Triangles.  We  often  have  to  construct 
triangles  of  various  shapes  and  sizes.  We  shall  first  con- 
sider the  following  case : 

Construct  a  triangle  having  its  sides  respectively  equal  to 
three  given  lines. 

Let  Z,  m,  n  be  the  given  lines. 

It  is  required  to  construct  a 
triangle  with  I,  m,  n  as  sides. 

Draw  a  line  with  the  ruler  and 
on  it  mark  off  with  the  compasses 

a  line  AB  equal  to  I. 

in 

It  is  more  nearly  accurate  to  do  this  

with  the  compasses  than  with  a  ruler. 

With  A  as  center  and  m  as  radius  draw  a  circle;  with 
B  as  center  and  n  as  radius  draw  another  circle  cutting 
the  first  at  C.  Draw  AC  and  BC. 

Then  because  AB=l,  AC=m,  and  BC=n  it  follows 
that  ABC  is  the  required  triangle. 

Show  why  it  is  not  necessary  to  draw  the  whole  circle  in  either  case. 
Teachers  should  informally  explain  to  the  students  the  methods 
commonly  used  in  lettering  a  line,  an  angle,  and  a  triangle. 

Exercise  2.    Triangles 

1.  Construct  a  triangle  with  sides  2  in.,  3  in.,  4  in. 

Construct  triangles  with  sides  as  follows : 

2.  3  in.,  4  in.,  5  in.  5.  -|  in.,  ^  in.,  1  in. 

3.  l£in.,  2  in.,  2^  in.  6.  2Jin.,  2J  in.,  2£  in. 

4.  1J  in.,  21  in.,  3  in.  7.  3  in.,  31  in.,  3J  in. 

In  schools  in  which  the  metric  system  is  taught  it  is  desirable  to 
use  the  system  in  this  work.  The  necessary  metric  measures  often 
will  be  found  on  protractors  such  as  the  one  shown  on  page  115. 


ANCIENT  INSTRUMENTS 


117 


uadrants  used  for  measuring  angles  hundreds 
of  years  ago.  German,  Italian,  and  Hindu  specimens, 


118 


GEOMETRY  OF  FOKM 


Isosceles  Triangle.  In  the  case  studied  on  page  116  we 
see  that  the  three  sides  need  not  all  be  equal.  If  two 
sides  are  equal  we  have  to  construct  an  isosceles  triangle. 

Construct  a  triangle  having  two  sides  each  equal  to  a  given 
line  and  the  base  equal  to  another  given  line. 

The  base  of  an  isosceles  triangle  is  always  taken  as  the  side  which 
is  not  equal  to  one  of  the  other  sides. 

Let  AB  be  the  given  base  and  let  I 
be  the  given  line. 

Then  with  center  A  and  radius  I 
draw  a  circle,  and  with  center  B  and 
radius  I  draw  another  circle,  or  pref- 
erably only  an  arc  in  each  case. 

Let  the  two  arcs  or  the  two  circles 
intersect  at  the  point  C. 

Then  ABC  is  the  triangle  required. 


l> 


Equilateral  Triangle.  From  the  preceding  case  we  see 
that  if  the  base  is  equal  to  each  of  the  other  sides,  we 
shall  have  an  equilateral  triangle. 

Exercise  3.    Isosceles  and  Equilateral  Triangles 

1.  In  making  a  pattern  for  the  tiles  used  in  the  floor 
shown  below  it  is  necessary  to  draw  an  equilateral  triangle 
of  side  1  in.  Draw  such  a  triangle. 


Construct  isosceles  triangles  with  bases 
1  in.  and  equal  sides  as  follows  : 

2.  fin.       3.  -Jin.       4.  1-J  in.         5.  2  in. 


rTTTTTTl 
YYYYYTY 

rYYYYYY1! 


6. 


Construct  equilateral  triangles  with  sides  as  follows : 

7.  4-  in.       8.  -Jin.       9.  1A  in.        10.  II  in.        11.  24  in. 

4  O  O  °  ~ 


ANCIENT  INSTRUMENTS 


119 


curious  illustration 
from  an  Italian  v>or^  of  the  seventeenth  century 
showing  the  use  of  the  ancient  quadrant. 
The  distance  was  required 


for  the  purpose  of  properly  fixing  the  guns. 

The  computations  may  be  made 

in  "various  •ways. 


120  GEOMETRY  OF  FORM 

12.  Cut  three  isosceles  triangles  of  different  shapes  from 
paper  and  fold  each  through  the  middle  so  that  one  of  the 
equal  sides  lies  exactly  on  the  other.    What  inference  can 
you  make  as  to  the  equality  or  inequality  of  the  angles 
which  are  opposite  the  equal  sides?    Write  the  statement 
as  follows :  In  an  isosceles  triangle  the  angles  opposite  the 
equal  sides  are  equal. 

13.  Draw  three  equilateral  triangles  of  different  sizes. 
With    a   protractor   measure    each  angle  in   each  of  the 
triangles.   What  inference  can  you  make  as  to  the  number 
of  degrees  in  each  angle  ?  Write  the  statement,  beginning 
as  follows:   The  number  of  degrees  in  each  angle  of,  etc. 

14.  From  Ex.  13  what  inference  can  you  make  as  to 
the  number  of  degrees  in  the  sum  of  the  three  angles  of 
an  equilateral  triangle  ?   This  is  the  same  as  the  number 
of  degrees  in  how  many  right  angles  ? 

15.  Draw  three  triangles  of  various  shapes  and  investi- 
gate for  each  the  conclusion  drawn  in  Ex.  14.    This  is 
most  easily  done  by  cutting 

them  from  paper  and  then 

cutting  off  the  three  angles 

in  each  case  and  fitting  them     A  B         x  Y 

together.     Write  the  statement,  beginning  as  follows:    In 

any  triangle  the  sum  of  the  three  angles  is  equal  to,  etc. 

16.  From  the  truth  discovered  in  Ex.  15,  find  the  third 
angle  of  a  triangle  in  which  two  angles  are  75°  and  45°. 

17.  In  a  certain  right  triangle  one  acute  angle  is  30°. 
How  many  degrees  are  there  in  the  other  acute  angle  ? 

In  this  work  the  student  is  led  to  discover  by  experiment 
various  important  propositions  to  be  proved  later  in  his  work  in 
geometry.  Teachers  may  occasionally  find  it  advantageous  to 
develop  simple  proofs  in  connection  with  this  intuitional  treatment. 


PEBPENDICULARS 


Perpendicular.    A  line  which  makes  a  right  angle  with 
another  line  is  said  to  be  perpendicular  to  that  line. 

One  of  the  best  practical  methods  of  constructing  a 
line  perpendicular  to  a  given  line  and  passing  through  a 
given  point  is  shown 
in  this  illustration. 

Place  a  right  tri- 
angle AB  C  so  that  BC 
lies  along  the  given 
line.  Lay  a  straight- 
edge or  ruler  along 
AC,  as  in  the  left-hand  figure.  Since  you  wish  the  per- 
pendicular line,  or  perpendicular,  to  pass  through  the  point 
P,  slide  the  triangle  along  MN  until  AB  passes  through  the 
point  P,  as  shown  in  the  right-hand  figure.  Then  draw 
a  line  along  AB,  and  it  will  be  perpendicular  to  the  line 
XY  and  will  pass  through,  the  point  P. 

Exercise  4.   Perpendiculars 

1.  Draw  a  line  XY  and  mark  a  point  P  about  -|in. 
below  it.    Through  P  construct  a  line  perpendicular  to  XY, 
by  the  above  method. 

2.  Through  a  point  P  on  the  line  XY  construct  a  line 
perpendicular  to  XY,  by  the  above  method. 

3.  Construct  a  right  triangle  in  which  the  two  shorter 
sides  shall  be  1|-  in.  and  2  in. 

4.  Construct  a  square  having  its  side  2  in. 

5.  Draw  a  picture  showing  how  two  carpenter's  squares 
can  be  tested  by  standing  them  on  any  flat  surface  with 
two  edges  coinciding  and  two  other  edges  extending  in 
opposite  directions. 


122  GEOMETRY  OF  FORM 

Other  Methods  of  Constructing  Perpendiculars.   There  are 
other  convenient  methods  of  constructing  perpendiculars. 

From  a  given  point  on  a  given  straight  line  construct  a 
perpendicular  to  the  line. 

Let  AB  be  the  given  line  and  P  be 
the  given  point. 

With  P  as  center  and  with  any  con- 
venient radius  draw  arcs  intersecting  A  I  \ 


-B 


A. — P" 

AB  at  X  and  Y.  u      * 

With  X  as  center  and  XY  as  radius  draw  a  circle,  and 

with  Y  as  center  and  the  same  radius  draw  another  circle, 

and  call  one  intersection  of  the  circles  C.  p 

With  a  ruler  draw  a  line  from  P  to  (7. 

From  a  given  point   outside   a  given 
straight  line  construct  a  perpendicular  to      3" 
the  line. 

Let  AB  be  the  given  line  and  P  be 
the  given  point.     How  are  the  points 
X  and  Y  fixed  ?   Then  how  is  the  point  C  fixed  ?    Draw 
the  perpendicular  PC. 

Exercise  5.  Perpendiculars 

1.  In  making  a  pattern  for  a  tiled  floor-  like  the  one 
here  shown  it  becomes  necessary  to  draw  a 

square  1  in.  on   a  side.    Construct  such   a 
square,  using  the  first  of  the  above  methods. 

2.  Construct  a  rectangle  as  in  Ex.  1,  using 
the  second  of  the  above  methods. 

3.  Given  two  points  on  a  given  line,  construct  perpen- 
diculars to  the  line  from  each  of  them. 


ANCIENT  INSTRUMENTS 


123 


arly  leveling  instruments^  wit  A  a  picture  from  a 
published  in  1624  showing  their  use. 


124 


GEOMETRY  OF  FORM 


Bisecting  a  Line.  To  divide  a  line  into  two  equal  parts 
is  to  bisect  it.  In  constructing  the  common  figures  we  often 
have  to  bisect  a  line.  We  can  bisect  a  line  ^x> 

roughly  by  measuring  it  with  a  ruler, 
but  for  accurate  work  we  have  a  much 
better  method. 


Bisect  a  given  line.  M 

Let  AB  be  the  given  line.  What  is 
now  required  ?  With  A  and  B  as  centers 
and  with  radius  greater  than  \AB  draw  "^" 

arcs.     The  most  convenient  radius  is  usually  AB  itself. 

Call  the  points  of  intersection  X  and  Y.  Draw  the 
straight  line  XY,  and  call  the  point  where  it  cuts  the 
given  line  M. 

Then  XY  bisects  AB  at  M. 

This  is  much  more  nearly  accurate  than  it  is  to  measure  the  line 
with  a  ruler  and  then  take  half  the  length. 

Bisecting  an  Angle.  To  draw  a  line  from  the  vertex  of 
an  angle  dividing  it  into  two  equal  angles  is  to  bisect  it. 

Bisect  a  given  angle. 

Let  AOB  be  the  given  angle. 

What  is  now  required  ? 

With  0  as  center  and  with  any  con- 
venient radius  draw  an  arc  cutting  OA 
at  X  and  OB  at  Y. 

With  X  and  Y  respectively  as  centers  and  with  a  radius 
greater  than  half  the  distance  from  X  to  Y  draw  arcs  and 
call  their  point  of  intersection  P.  Draw  OP. 

Then  OP  is  the  required  bisector. 

This  is  much  more  nearly  accurate  than  it  is  to  measure  the  angle 
with  a  protractor  and  then  take  half  the  number  of  degrees. 


SIMPLE  CONSTRUCTIONS  125 

Exercise  6.   Simple  Constructions 

1.  Draw  a  line  4.5  in.  long.    Bisect  this  line  with  ruler 
and  compasses.  Check  the  construction  by  folding  the  paper 
at  the  point  of  bisection,  making  a  fine  pinhole  through  one 
end  of  the  line  to  see  if  it  strikes  the  other  end. 

2.  Construct  a  triangle  having  two  of  its  sides  3  in.,  the 
third  side  being  less  than  6  in. 

3.  Construct  a  triangle  having  its  sides  respectively 
2  in.,  2.5  in.,  and  3  in. 

4.  Draw  a  line  4  in.  long,  and  at  a  point  1  in.  from 
either  end  construct  a  perpendicular  to  the  line. 

5.  Is  it  possible  to  construct  a  triangle  having  its  sides 
respectively  3  in.,  2  in.,  and  1  in.  ?    If  not,  what  is  there 
in  the  general  nature  of  these  lengths  which  makes  such 
a  triangle  impossible  ? 

6.  With   a  protractor  draw  an   angle   of   35°.    Bisect 
this  angle  and  check  the  work  with  the  protractor. 

To  draw  the  angle  of  35°  draw  a  line,  mark  a  point  O  upon  it, 
lay  the  hypotenuse  of  a  triangular  protractor  on  it,  sliding  it  down 
slightly  so  that  the  center  of  the  circle  rests  on  O.  Lay  a  ruler  on 
the  protractor  from  0  along  the  line  of  35°  and  mark  a  point  on  the 
paper.  Remove  the  protractor  and  draw  a  line  from  0  to  the  point. 

Construct  the  triangles  ivhose  sides  are  as  follows  and  bisect 
all  three  angles  of  each  triangle: 

7.  4  in.,  3  in.,  41  in.  10.  3|-  in.,  4^  in.,  5^  in. 

8.  5  in.,  7  in.,  8  in.  11.  7-|  in.,  4-|  in.,  5  in. 

9.  6  in.,  3  in.,  5  in.  12.   3-|  in.,  3^  in.,  3-|  in. 

13.  In  Exs.  7-12  what  do  you  observe  as  to  the  way 
in  which  the  three  bisectors  meet  ?  Write  a  statement  of 
your  conclusion,  beginning  .as  follows:  The  bisectors  of 
the  three  angles  of  a  triangle,  etc. 


126  GEOMETRY  OF  FORM 

Constructing  an  Angle  equal  to  a  Given  Angle.  In  copy- 
ing figures  we  often  have  to  construct  an  angle  equal  to 
a  given  angle.  This  leads  to  the  following  construction : 

From  a  given  point  on  a  given  line  construct  a  line  which 
shall  make  with  the  given  line  an  angle  equal  to  a  given  angle. 


15 Q 


Let  P  be  the  given  point  on  the  given  line  PQ  and  let 
angle  AOB  be  the  given  angle. 

What  is  now  required  ? 

With  0  as  center  and  with  any  radius  draw  an  arc  cut- 
ting OA  at  C  and  OB  at  D. 

\Vith  P  as  center  and  with  OC  as  radius  draw  an  arc 
cutting  PQ  at  M. 

With  M  as  center  and  with  the  straight  line  joining  C 
and  D  as  radius  draw  an  arc  cutting  the  arc  just  drawn  at 
JV,  and  draw  PN. 

Then  the  angle  MPN  is  the  required  angle. 

Exercise  7.    Simple  Constructions 

Construct  triangles  with  sides  as  follows  and  bisect  all 
three  of  the  sides  of  each  triangle : 

1.  5  in.,  6  in.,  Tin.  4.  3  in.,  3^  in.,  4J  in. 

2.  4  in.,  4  in.,  7  in.  5.  2J  in.,  4  in.,  4  in. 

3.  3^  in.,  4£  in.,  7-|  in.  6.  3£  in.,  3^  in.,  4  in. 

Interesting  figures  may  be  formed  by  connecting  the  points  of 
bisection  and  shading  in  various  ways  the  parts  thus  formed. 


SIMPLE  CONSTRUCTIONS  127 

7.  Construct  a  triangle  ABC  with  AB  =  1  in.,  AC—\\  in., 
angle  A  =  30°,  and  then  construct  another  triangle  XYZ 
with  XF=1  in.,   XZ=lJin.,   angle   X=  30°.     Are   the 
triangles  ABC  and  XYZ  congruent? 

8.  From    Ex.  7   write    a   complete    statement    of    the 
truth  inferred,  beginning  as  follows:    Two  triangles  are 
congruent  if  two  sides  and  the  included  angle  of  one  are 
respectively  equal  to,  etc. 

9.  Construct  a  triangle  ABC  in  which  angle  A  —  30°, 
angle  B  =  60°,  AB  =  1^-  in.,  and  then  construct  another 
triangle   XYZ  in  which    angle    X=  30°,    angle    r=60°, 
-YF=l^in.     Are   these   triangles   congruent?     What   is 
the  reason  ?     Write  a  complete   statement  of  the   truth 
inferred,  as  in  Ex.  8. 

10.  Construct   a   triangle    ABC   in    which   AB  =  1  in., 
2?  (7=1^  in.,  CL4=l|-in.,  and  then  construct  another  triangle 
XYZ  in  which  XY=  1  in.,  YZ=l^  in.,  ZX=  1£  in.    Are 
these  triangles  congruent?     Write  a  complete  statement 
of  the  truth  inferred,  as  in  Ex.  8. 

11.  Construct  a  triangle  with  angles  30°,  60°,  and  90°, 
and  another  triangle  with  sides  twice  as  long  but  with 
angles   the    same.    Are  these  triangles   congruent?    Are 
triangles  in  general  congruent  if  the  angles  of  one  are 
respectively  equal  to  the  angles  of  the  other  ? 

12.  As  in  Ex.  11,  construct  two  triangles  with  angles  45°, 
45°,  and  90°,  one  with  sides  three  times  as  long  as  the  other. 

13.  Try  to  construct  a  triangle  with  angles  45°,  60°, 
and  90°.    If  you  have  any  difficulty  in  making  the  con- 
struction, write  a  statement  of  the  cause. 

14.  Try  to  construct  a  triangle  with  angles  45°,  45°,  and 
100°.  If  you  have  any  difficulty  in  making  the  construction, 
write  a  statement  of  the  cause. 


128 


GEOMETRY  OF  FOKM 


Parallel  Lines.  One  of  the  most  common  constructions 
in  making  architectural  and  mechanical  drawings  is  to  draw 
one  line  parallel  to  another  line.  For  „  ^ 

practical  purposes  one  of  the  best  plans 
is  to  place  a  wooden   or  celluloid  tri- 
angle ABC  with   one  side  BC  on  the 
given   line,   lay  a  ruler  along   another 
side  AB,  and  then  slide  the  triangle  along  the  ruler  to  the 
position  A'B'C'  (read  .4-prime,  J5-prime,   (7-prime).    Then 
B'C'  is  parallel  to  BC. 

A  triangular  protractor  like  the  one  shown  on  page  115  of  this 
book  may  be  used  for  the  above  purpose. 

Draftsmen  in  offices  of  architects  or  in  machine  shops 
often  use  a  T-square  as  here  shown.  As  the  part  MN  slides 
along  the  edge  CD  of  a  draw- 
ing board,  the  part  OP  moves 
parallel  to  its  original  position. 
Drawing  EF  and  sliding  the 
T-square  along,  we  can  easily 
draw  lines  parallel  to  EF.  A 
second  T-square  may  slide  along 
BD  if  the  board  is  rectangular, 
and  thus  lines  can  be  drawn  perpendicular  to  the  line  EF 
or  to  any  lines  parallel  to  it. 

When  the  lines  are  very  long,  this  is  the  best  method. 

Draftsmen  also  use  a  parallel  ruler  like  the  one  here 
shown.  They  also  use  a  cylindric  ruler,  rolling  it  along 

the  paper  as  a  guide  for  r-r — i 

parallel   lines.     In   gen-  VJ  /? 

eral,  however,  the  plan    J/ II 

of  sliding  a  triangle  along 

a  ruler  is  one  of  the  simplest  and  at  the  same  time  is 

accurate.    It  should  be  used  in  the  exercises  which  follow. 


D 


ANCIENT  INSTRUMENTS 


129 


Early  uses  of  geometry  in  studying  the  stars, 
ofe,  an  astrolabe  used  in  measuring  the  angles  of  stars  abo'Ve  the 
orizon.  "Below,  an  ancient  Hindu  bronze  sphere  of  the  AeaTens,  -with 
stars  inlaid  in  silver. 


130 


GEOMETRY  OF  FORM 


L     M    N     O 


Dividing  a  Line.    We  often  need  to  divide  a  line  into  a 
given  number  of  equal  parts;  that  is,  to  solve  this  problem: 

Divide  a  given  line  into  any  given  number  of  equal  parts. 

Let  AB  be  the  given  line,  and 
let  it  be  required  to  divide  AB 
into  five  equal  parts. 

Draw  any  line  from  A,  as  AX. 
Mark  off  on  AX  with  the  com- 
passes any  five  equal  lengths  AP,  PQ,  QR,  RS,  and  ST. 

Draw  TB,  and  then,  by  sliding  a  triangle  along  a  ruler, 
draw  SO,  RN,  QM,  and  PL  parallel  to  TB. 

Then  AB  is  divided  into  five  equal  parts,  AL,  LM,  MN, 
NO,  and  OB. 

The  material  for  another  very  simple  method  may  be 
easily  prepared  by  the  student.  Let  him  rule  a  large 
sheet  of  paper  with 
several  parallel  lines 
at  equal  intervals, 
and  number  these 
lines  as  shown  on 
the  edge.  If  it  is 
desired  to  divide  the 
line  AB  into  five 
equal  parts,  place  the 
paper  on  which  AB 
is  drawn  over  the 
ruled  paper  so  that 
the  line  0  passes 
through  A  and  the 
line  5  through  B.  Lay  the  ruler  along  each  ruled  line  in 
turn  and  mark  each  point  of  division.  In  this  way  the 
four  required  points  of  division  may  be  accurately  found. 


SIMPLE  CONSTRUCTIONS  131 

Exercise  8.   Simple  Constructions 

1.  Draw  a  line  5  in.  long  and  divide  it  into  nine  equal 
parts  by  using  ruler,  triangle,  and  compasses. 

Construct  triangles  whose  sides  are  as  follows,  and  con- 
struct a  perpendicular  to  each  side  at  its  midpoint: 

2.  4-|  in.,  4^  in.,  5  in.  4.  5-^-  in.,  5^  in.,  6^  in. 

3.  3J  in.,  31  in.,  51  in.  5.  4  in.,  5J  in.,  6J  in. 

The  teacher  should  ask  for  the  inference  as  to  the  meeting  of  the 
three  perpendicular  bisectors  of  the  sides  of  a  triangle. 

6.  With  a  protractor  draw  an  angle  of  45°.   With  ruler 
and  compasses  bisect  this  angle.    Check  the  construction 
by  folding  the  paper ;  by  using  the  protractor. 

7.  Draw  any  triangle,  bisect  the   sides,  and  join  the 
points  of  bisection,  thus  forming  another  triangle.    With 
ruler  and  triangle  test  to  see  whether  the   sides  of  the 
small  triangle  are  parallel  to  those  of  the  large  triangle. 

8.  Repeat  Ex.  7  for  a  triangle  of  different  shape.    What 
general  law  do  you  infer  from  these  two  cases  ? 

v  9.  Draw  a  line  4^  in.  long  and  divide  it  into  seven 
equal  parts. 

''  10.  Construct  a  square  3  in.  on  a  side.  If  the  figure  is 
correctly  drawn,  the  two  diagonals  will  be  equal.  Check 
by  measuring  the  diagonals  with  the  compasses. 

If  such  words  as  diagonal  are  not  familiar  they  should  be  explained 
by  the  teacher  when  they  are  met.  It  is  desirable  to  avoid  formal 
definitions  at  this  time,  provided  the  students  use  the  terms  properly. 

11.  Construct  two  parallel  lines  and  draw  a  slanting 
line  cutting  these  lines  so  that  eight  oblique  angles  are 
formed.  Name  the  various  pairs  of  angles  in  the  figure 
that  appear  to  be  equal. 


132 


GEOMETRY  OF  FORM 


Geometric  Patterns.  By  the  aid  of  the  constructions 
described  on  pages  116-130  it  is  possible  to  construct  a 
large  number  of  useful  and  interesting  patterns,  designs 
for  decorations,  and  plans  for  buildings  or  gardens. 

To  secure  the  best  results  in  this  work  the  pencil 
should  be  sharpened  to  a  fine  point  and  should  contain 
rather  hard  lead,  and  the  lines  should  be  drawn  very  fine. 


Exercise  9.    Geometric  Patterns 

1.  By  the   use   of   compasses   and  ruler   construct  the 
following  figures : 


The  lines  made  of  short  dashes  show  how  to  fix  the  points  needed 
in  drawing  a  figure,  and  they  should  be  erased  after  the  figure  is 
completed  unless  the  teacher  directs  that  they  be  retained  to  show 
how  the  construction  was  made. 

2.  By  the  use  of  compasses  and  ruler  construct  the 
following  figures : 


It  is  apparent  from  the  figures  in  Exs.  1  and  2  that  the  radius  of 
the  circle  may  be  used  in  drawing  arcs  which  shall  divide  the  circle 
into  six  equal  parts  by  simply  stepping  round  it. 


GEOMETRIC  PATTERNS 


133 


3.  By  the  use  of  compasses  and  ruler  construct  the 
following  figures,  shading  such  parts  as  will  make  a 
pleasing  design  in  each  case : 


4.  By  the  use  of  compasses  and  ruler  construct  the 
following  figures,  shading  such  parts  as  will  make  a 
pleasing  design  in  each  case : 


5.  By  the   use   of   compasses   and   ruler   construct  the 
following  figures : 


In  such  figures  artistic  patterns  may  be  made  by  coloring  portions 
of  the  drawings.  In  this  way  designs  are  made  for  stained-glass 
windows,  for  oilcloths,  for  colored  tiles,  and  for  other  decorations. 


134 


GEOMETKY  OF  FORM 


6.  By  the   use   of   compasses   and  ruler  construct   the 
following  figures,  leaving  the   dotted  construction   lines: 


As  stated  on  page  133,  artistic  patterns  may  be  made  by  coloring 
various  parts  of  these  drawings.  Interesting  effects  are  also  pro- 
duced in  black  and  white,  as  in  the  designs  in  Ex.  9  on  page  135. 

7.  Draw  a  line  1^  in.  long  and  divide  it  into  eighths  of  an 
inch,  using  the  ruler.    Then  with  the  compasses  construct 
this  figure. 

It  is  easily  shown,  when  we  come 
to  the  measurement  of  the  circle,  that 
these  two  curve  lines  divide  the  space 
inclosed  by  the  circle  into  parts  that 
are  exactly  equal  in  area. 

By  continuing  each  semicircle  to 
make  a  complete  circle  another  inter- 
esting figure  is  formed.  Other  similar 
designs  are  easily  invented,  and  stu- 
dents should  be  encouraged  to  make 
such  original  designs. 

8.  In  planning  a  Gothic  window  this  drawing  is  needed. 
The  arc  BC  is  drawn  with  A  as  center 

and  AB  as  radius.  The  small  arches 
are  drawn  with  A,  D,  and  B  as  centers 
and  AD  as  radius.  The  center  P  is 
found  by  using  A  and  B  as  centers  and 
AE  as  radius.  How  may  the  points  Z>, 
E,  and  F  be  found  ?  Draw  the  figure.  A 


GEOMETRIC  PATTERNS 


135 


9.  Copy  each  of  the  following  designs,  enlarging  each 
to  twice  the  size  shown  on  this  page : 


This  example  and  the  following  examples  on  this  page  may  be 
omitted  by  the  class  at  the  discretion  of  the  teacher  if  there  is  not 
enough  time  for  such  work  in  geometric  drawing. 

10.  This   figure    shows   a  piece    of   inlaid  work   in   an 
Italian  church.    Construct  a  design  of  this  general  nature, 
changing  it  to  suit  your  taste. 

Construct   the    figures   as   accu- 
rately as  you  can. 

11.  Construct  a  design  for  par- 
quetry flooring,  using  only  com- 
binations of  squares. 

12.  Repeat  Ex.  11,  using  com- 
binations of  squares  and  equilat- 
eral triangles. 

13.  Repeat  Ex.  11,  using  combinations  of  squares,  rec- 
tangles, and  equilateral  triangles. 

14.  Construct  a  design  for  a  geometric  pattern  for  lino- 
leum, using  only  combinations  of  circles  and  squares. 

15.  Repeat  Ex.  14,  using  only  combinations  of  circles 
and  equilateral  triangles. 

16.  Repeat  Ex.  14,  using  only  combinations  of  circles, 
squares,  and  equilateral  triangles. 


136 


GEOMETRY  OF  FOKM 


Drawing  to  Scale.  The  ability  to  understand  drawings, 
maps,  and  other  graphic  representations  depends  in  part 
upon  knowing  how  to  draw  to  scale. 

Thus,  if  your  schoolroom  is  30  ft.  long  and  20  ft.  wide, 
and  you  make  a  floor  plan  3  in.  long  and  2  in.  wide,  you 
draw  the  plan  to  scale,  1  in.  representing  10  ft.  We  indi- 
cate this  by  writing:  "Scale,  1  in.  =  10  ft."  We  may  also 
write  this:  "Scale,  1  in.  =  120  in.,"  or  "Scale  ^5,"  We 
often  write  1'  for  1  ft.  and  1"  for  1  in.,  so  that  the  scale 
may  also  be  indicated  as  1"  =  10'. 

The  following  shows  a  line  AS  drawn  to  different  scales : 

^J.  £j 


The  line  AB  drawn  to  the  scale 
The  line  AB  drawn  to  the  scale 
The  line  AB  drawn  to  the  scale 


The  figures  shown  below  illustrate  the  drawing  of  a 
rectangle  to  scale.  In  this  case  the  lower  rectangle  is  a 
drawing  of  the  upper 
one  to  the  scale  '  -^-,  or 
1  to  2,  or  1"  to  2". 

Notice  that  the  area  of 
the  lower  rectangle  is  only 
£  that  of  the  upper  one. 
When  we  draw  to  the  scale  -| 


we  mean  that  the  length  of  every  line  is  ^  the 
length  of  the  corresponding  line  in  the  original. 
Whatever  the  shape  of  the  figure,  the  area  will 
then  be  ^  the  area  of  the  original  figure. 

Maps  are  figures  drawn  to  scale.  The  scale  is  usually 
stated  on  the  map,  as  you  will  see  in  any  geography. 
The  scale  used  on  a  map  is  often  expressed  by  means  of 
a  line'  divided  to  represent  miles,  and  sometimes  by  such 
a  statement  as  that  1  in.  =  100  mi. 


1ST 


Exercise  10.   Drawing  to  Scale 

1.  Measure  the  cover  of  this  book.    Draw  the  outline 
to  the  scale  ^. 

This  means  that  the  four  edges  are  to  be  drawn  to  form  a  rec- 
tangle like  the  front  cover,  with  no  decorations. 

2.  Measure   the   top   of  your  desk.     Draw  a  plan  to 
the  scale  ^. 

3.  If  a  line  1  in.  long  in  a  drawing  represents  a  dis- 
tance of  8  ft.,  what  distance  is  represented  by  a  line  3f  in. 
long  ?    by  a  line  4|-  in.  long  ?    by  a  line  1.5  in.  long  ? 

4.  If   the   scale    is    1  in.    to   1  ft,   what  distance   on   a 
drawing  will  represent  6  ft.  3  in.  in  the  object  drawn  ? 

5.  A  drawing  of  a  rectangular  floor  20  ft.  by  28  ft.  is 
5  in.  by  7  in.    What  scale  was  used  ? 

6.  A  farmer  plotted  his  farm  as  here  shown,  using  the 
scale  of  1  in.  to  40  rd.    Find  the  dimensions  of  each  plot. 


WHKAT 


OATS 


CORN 


PASTURE 


WOOD  LOT 


!    O  [DWELLING 

!    §   j      AND 
i    S        BARNS 


7.  A  plan  of  a  rectangular  school  garden  is  drawn  to 
the  scale  of  1  in.  to  2  ft.  6  in.     The  plan  is  18  in.  long 
and  12^ in.  wide.   What  are  the  dimensions  of  the  garden? 

8.  The  infield  of  a  baseball  diamond  is  90ft.  square. 
Draw  a  plan  to  the  scale  of  1  in.  to  20  ft. 


138 


GEOMETRY  OF  FORM 


9.  The  field  of  play  of  a  football  field  is  300  ft.  long 
and  160  ft.  wide.  Lines  parallel  to  the  ends  of  the  field 
are  drawn  at  intervals  of  5  yd.,  and  the  goals,  18  ft.  6  in. 
wide,  are  placed  at  the  middle  of  the  ends  of  the  field. 
Draw  a  plan  to  the  'scale  of  1  in.  to  60  ft.  and  indicate  the 
position  of  the  goals  and  of  the  5-yard  lines. 

10.  A  double  tennis  court  is  78  ft.  long  and  36  ft.  wide. 
Lines  are  drawn  parallel  to  the  longer  sides  and  4  ft.  6  in. 
from  them,  and  the  service  lines  are  parallel  to  the  ends 
and  18  ft.  from  them.     The  net  is  halfway  between  the 
ends.    Draw  a  plan  to  any  convenient  scale. 

11.  The  drawing   here    shown    is    the   floor  plan   of   a 
certain  type  of  barn.     Determine  the  scale  to  which  the 


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plan  is  drawn,  find  the  width  of  the  driveway  in  the  barn, 
the  width  bf  each  horse  stall,  the  width  of  each  cattle  stall, 
and  the  dimensions  of  the  box  stall  and  the  feed  room. 


DRAWING  TO  SCALE 


139. 


12.  A  class  in  domestic  science  drew  a  plan  for  a  model 
kitchen  in  an  apartment  house,  using  the  scale  -^-.   If  the 
plan  is   3  in.  long  and   2  in.   wide,   what   are  the   actual 
dimensions  of  the  kitchen  ? 

13.  A  drawing  was  made  of  a  lamp  screen  20-£  in.  high.. 
The  drawing  being  2^- in.  high,  what  scale  was  used? 

14.  The    drawing    below   is    the   plan    for    a   concrete 
bungalow.     Find  the  scale  used  in  drawing  the  plan. 


15.  In  Ex.  14  find  the  dimensions  of  the  living  roon\ 
dining  room,  and  smaller  bedroom  including  wardrobe. 


140  GEOMETRY  OF  FORM 

Accurate  Proportions.  Suppose  that  you  measure  a  rec- 
tangular room  and.  find  it  to  be  20  ft.  long  and  16  ft.  wide, 
and  suppose  that  you  measure  a  drawing  of  the  room  and 
find  it  to  be"  10  in.  long  and  6  in.  wide.  You  would  con- 
clude that  the  drawing  is  not  a  good  one,  because  the 
width  should  be,  as  in  the  room,  |-  of  the  length. 

An  accurate  drawing  or  picture  must  maintain  the  pro- 
portions  of  the  object. 

That  is,  if  the  width  of  the  object  is  ^  of  the  length,  the 
width  of  the  object  shown  in  the  drawing  must  be  ^  of 
the  length  in  the  drawing ;  if  the  width  of  the  object  is  •§• 
of  the  length  in  one  case,  it  must  be  -g-  of  the  length  in 
the  other  case ;  and  so  on  for  other  proportions. 

It  is  better  at  this  time  to  explain  informally  the  meaning  of 
proportion,  as  is  done  above.  A  more  formal  explanation  of  the 
subject  of  proportion  is  given  later  in  the  book  when  it  is  needed. 

Exercise  11.   Accurate  Proportions 

1.  A  house  is  36  ft.  high  and  the  garage  is  20  ft.  high. 
If  the  house  is  represented  in  a  drawing  as  18  in.  high, 
how  high  should  the  drawing  of  the  garage  be  ? 

In  all  such  cases  the  objects  are  supposed  to  be  at  approximately 
the  same  distance  from  the  eye,  so  that  the  element  of  perspective 
does  not  enter. 

2.  A  landscape  gardener  is  drawing  to  scale  a  plan  for  a 
rectangular  flower  garden  18  ft.  long  and  14  ft.  wide.    In 
the  drawing  the  length  is  represented  by  6^  in.    By  what 
should  the  width  be  represented  ? 

3.  Draw  a  right  triangle  whose  sides  are  3  in.,  4  in.,  and 
5  in.  respectively,  and  draw  another  right  triangle  of  the 
same  shape  but  with  the  hypotenuse  l^in.  long. 


SIMILARITY  OF  SHAPE 


141 


Similarity  of  Shape.  As  we  have  already  seen,  it  is 
frequently  necessary  to  draw  a  figure  of  the  same  shape 
as  another  one,  but  not  of  the 
same  size.  For  example,  an 
architect  or  a  map  drawer  may 
reduce  the  original  by  using 
a  small  scale,  but  if  we  are 
making  a  drawing  of  a  small 
object  seen  through  a  micro- 
scope we  use  a  large  scale.  But  whether 
the  drawing  reduces  or  enlarges  the 
original,  the  shape  remains  the  same. 

Figures   which   have    the    same    shape 
are   said  to  be   similar. 

For  example,   here   are  two   drawings 
of  a  hand  mirror.    In  outline  each  drawing  is  similar  to 
the  mirror  itself,  and  each  is  also  similar  to  the  other. 

Figures  which  are  similar  to  the  same  figure  are  similar 
to  each  other. 

Two  maps  of  a  state  are  not  only  similar  in  outline  to  the  state 
itself,  but  each  is  similar  in  outline  to  the  other. 

Exercise  12.   Similarity  of  Shape 

1.  Construct  three  equilateral  triangles  whose  sides  are 
respectively  2  in.,  3|-  in.,  and  5  in.    Are  they  similar  ? 

2.  Construct  three  rectangles,  the  first  being  1^-  in.  by 
2|-  in. ;  the  second,  3  in.  by  5  in. ;  and  the  third,  2  in.  by 
2-^-  in.    If  they  are  not  all  of  the  same  shape, 

discuss  the  exception. 

3.  Construct  a  right  triangle  of  the  same 
shape   as  this   triangle   but   twice  as  high, 


and  another  of  the  same  shape  but  three  times  as  high. 


142 


GEOMETRY  OF  H)KM 


Angles  in  Similar  Figures.  Here  are  two  similar  right 
triangles,  ABC  and  A'B'C',  and  in  each  triangle  a  perpen- 
dicular (jt?,  p'  respectively)  is  drawn  from  the  vertex  of 
the  right  angle  to  the  hypotenuse. 


B' 


Are  the  figures  still  similar  ?  Are  the  sides  proportional  ? 
What  can  be  said  as  to  the  corresponding  angles? 

This  brings  us  to  another  property  of  two  similar  figures, 
namely,  that  the  angles  of  one  are  equal  respectively  to  the 
angles  of  the  other.  That  is,  in  similar  figures,  corresponding 
lines  are  in  proportion  and  corresponding  angles  are  equal. 

A  close  approximation  to  similar  figures  may  be  seen  in  the  case 
of  moving  pictures.  The  large  picture  shown  on  the  screen  is  sub- 
stantially similar  to  the  small  picture  on  the  reel,  although  there  is 
some  distortion,  particularly  around  the  edges. 


Exercise  13.    Similar  Figures 

All  work  oral 

State  which  of  the  following  pairs  of  figures  are  necessarily 
similar  and  state  briefly  the  reasons  in  each  case: 

1.  Two  squares.  4.  Two  rectangles. 

2.  Two  triangles.  5.  Two  isosceles  triangles. 

3.  Two  circles.  6.  Two  equilateral  triangles. 

7.  State  whether  two  parallelograms,  each  side  of  one 
being  3  in.  and  each  side  of  the  other  being  4  in.,  must 
always  be  similar,  and  give  the  reason  for  your  answer. 


SIMILAR  FIGURES 


143 


Similar  Figures  in  Photographs.  If  you  have  ever  used 
a  plate  camera  you  have  seen  that  there  is  a  piece  of 
ground  glass  in 
the  back  and  that 
an  object  in  front 
of  the  camera  ap- 
pears inverted  on 
this  ground  glass. 
The  reason  is  clear, 
for  the  ray  of  light 
from  the  point  A  of  the  flower  passes  through  the  lens  of 
the  camera  and  strikes  the  plate  at  A.  That  is, 

On  a  photographic  plate  the  figure  is  similar  in  outline  to 
the  original,  but  is  inverted. 

There  is,  of  course,  a  slight  distortion  on  account  of  the  refrac- 
tion of  the  rays  of  light  in  passing  through  the  lens. 

If  the  camera  is  8  in.  long  and  the  object  is  16  in.  away 
from  the  lens  0,  an  object  5  in.  high  will  appear  as  2*  in. 
high  on  the  plate.    That 
is,  since  the  length  of  the  — '     "]5in 

B    _Jin>  ~-~ 16  in. 

camera,  B'O,   is   half  the     2*in-JI--~-  ~~~~o=~' 

distance  of  the  object  from 

0,  or  half  of  OB,  we  see  that  A'B',  the  height  of  the  object 

on  the  plate,  is  half  of  AB,  that  is,  half  the  real  height. 

Similarly,  if  the  length  of  the  camera  is  10  in.,  arid  the 
height  of  an  object  18  ft.  away  is  5  ft.,  we  can  easily  find 
the  height  of  the  object  on  the  plate  as  follows : 

Reducing  all  the  measurements  to  inches,  we  have 
18  ft.  =  18  X  12  in.,  and  5  ft.  =  5  x  12  in. 
10 


Then 


x  5  x  12  in.  =  2|  in. 


18  x  12 
The  teacher  is  advised  to  solve  this  on  the  blackboard. 


144  GEOMETRY  OF  FORM 

Exercise  14.   Similar  Figures  in  Photographs 

1.  A   man  5  ft.  8  in.  tall  stands  16  ft.  from  a  camera 
which   is    8  in.    long.     What   will   be   the   height   of   his 
photograph?     Explain   by   drawing  to   scale. 

2.  The  photograph  of  a  man  who  is  5  ft.  8  in.  tall  is 
6  in.  high,  and  the  camera  is  10  in.  long.    How  far  did  the 
man  stand  from  the-  camera  ? 

3.  If  a  boy's  face  is  8  ft.  from  a  camera  which  is  10  in. 
long,  the  height  of  the  photograph  of  his  face  is  what  pro- 
portion to  the  height  of  his  face?    If  he  places  his  hand 
2  ft.  nearer  the  camera,  the  length  of  the  photograph  of 
his  hand  is  what  proportion  to  the  length  of  his  hand  ? 

One  of  the  first  things  a  beginner  has  to  learn  in  using  a  camera 
is  that  objects  appear  distorted  unless  they  are  at  about  the  same 
distance  from  the  camera,  especially  if  they  are  relatively  near  to  it. 

4.  A  tree  photographed  by  a  4-inch  camera  at  a  distance 
of  10  ft.  appears  on  the  photograph  as  6  in.  high.     How 
high  is  the  tree? 

We  see  by  this  problem  that  heights  and  distances  can  often  be 
found  by  photography ;  and,  in  fact,  much  difficult  engineering  work 
is  now  done  with  the  aid  of  photographs. 

5.  A  camera  is  held  directly  in  front  of  the  middle  of 
a  door  and  at  a  distance  of  8  ft.  from  it.     The  door  is 
4  ft.  by  7  ft.  6  in.  and  the  length  of  the  camera  is  8  in. 
Find  the  dimensions  of  the  door  in  the  photograph. 

6.  A   10-inch   camera  is   placed    at   a  certain   distance 
from  a  tree   which   is   50  ft.   high,   and   a  boy   5  ft.   tall 
stands  between  the  tree  and  the  camera.    The  height  of 
the  boy  in  the  photograph  is  1-J-  in.,  and  the  height  of  the 
tree  8  in.    Find  the  distance  of  both  the  boy  and  the  tree 
from  the  camera. 


THE  PANTOGRAPH  145 

The  Pantograph.  Probably  you  have  seen  an  instru- 
ment which  is  extensively  used  by  architects,  draftsmen, 
designers,  and  map  makers  in  drawing  plane  figures  simi- 
lar to  other  plane  figures.  It  usually  consists  of  four  bars 
parallel  in  pairs,  and  is  known  as  a  pantograph. 

In  explaining  the  pantograph  it  becomes  necessary  to 
speak  of  the  ratio  of  two  lines.  By  the  ratio  of  2  ft. 
to  5  ft.  is  meant  the  quotient  2  ft.  -f-  5  ft.,  or  |-,  and  by 
the  ratio  of  ^  in.  to  3^  in.  is  meant  \  in.  -s-  3^  in.,  or  -^. 
Likewise,  by  the  ratio  of  a  line  AB  to  a  line  CD  is  meant 
the  quotient  found  by  dividing  the  length  of  the  line  AS 
by  the  length  of  CD.  This  ratio  is  written  AB/CD,  or 
AS:  CD.  If  AB  is  half  CD,  then  AB:CD  =  \.  This  is 
read  "the  ratio  of  AB  to  CD  is  equal  to  one  half." 

In  the  figure  the  bars  are  adjustable  at  B  and  E.  The 
end  A  is  fixed,  that  is,  it  remains  in  the  same  place  while 
the  pantograph  is  c 

being  used.  A  trac- 
ing point  is  placed 
at  T  and  a  pencil  at 
P,  and  BP  and  PE 
are  so  adjusted  as 
to  form  a  parallelogram  PECB  such  that  any  required 
ratio  AB-.AC  is  equal  to  CE-.CT.  Then  as  the  tracer  T 
traces  a  given  figure,  the  pencil  P  draws  a  similar 
figure.  If  the  given  figure  is  to  be  enlarged  instead  of 
reduced,  the  pencil  and  the  tracing  point  are  interchanged. 

This  discussion  of  the  pantograph  has  little  value  unless  the  in- 
strument is  actually  used  by  the  students.  A  fairly  good  one  can  be 
made  of  heavy  cardboard  or  of  strips  of  wood,  and  school-supply 
houses  will  furnish  ,a  school  with  the  instrument  at  a  low  cost. 

A  simple  pantograph  can  be  made  by  fastening  a  rubber  elastic 
at  one  end,  sticking  a  pencil  point  through  the  other  end,  and  placing 
a  pin  for  a  tracer  anywhere  along  the  band. 


146 


GEOMETKY  OF  FORM 


Exercise  15.    The  Pantograph 

1.  Draw  a  'plan  of  your  schoolroom  to  scale  and  then 
enlarge  it  to  twice  the  size  with  the  aid  of  a  pantograph. 

This  exercise  should  be  omitted  in  case  the  school  is  not  supplied 
with  a  pantograph. 

2.  Find    the   map    of    your 
state  in  a  geography  and  re- 
duce   it   to    half   the    size    by 
using  a  pantograph. 

3.  By  using    a   pantograph 
reduce    the   size   of   this   plan 
of  a  cottage  to  two  thirds  its 
present  size. 

This  can  be  done  by  laying  this 
page  flat  on  the  drawing  board 
while  someone  holds  the  book.  It  is 
better,  however,  to  copy  the  plan  on 
paper  and  use  the  pantograph  with 
the  drawing.  It  is  desired  that  the 
•student  should  use  the  pantograph 
•a  few  times  in  connection"  with 
Various  kinds  of  work  in  which  it  is  really  used  in  practical  life. 

4.  By  using   a  pantograph   enlarge   this    sketch   for   a 
child's  coat  to  three  times  the  given  size. 

In  addition  to  this,  other  similar  drawings  should  be 
made  and  then  enlarged.  Of  late  the  pantograph  has 
come  into  extensive  use  by  dressmakers  for  the  purpose 
of  enlarging  designs  of  this  kind. 

5.  Draw  a  sketch  of  any  object  in  the  room  and  reduce 
the  sketch  to  one  third  its  size  by  using,  a  pantograph. 

6.  Draw  a  sketch  of  a  tree  near  the  school  and  enlarge 
the  sketch  to  five  times  its  size  by  using  a  pantograph. 


FIRST  FLOOR  FMN 


SYMMETRY 


14T 


Symmetry.  If  we  place  a  drop  of  ink  on  a  piece  of 
paper  and  at  once  fold  the  paper  so  as  to  spread  the  ink, 
we  shall  often  find  curious  and  interest- 
ing forms  frequently  resembling  flowers, 
leaves,  or  butterflies.  These  forms  are 
even  more  interesting  if  we  use  a  drop 
of  black  ink  and  a  drop  of  red  ink. 
The  interest  in  such  figures  comes  from 
the  fact  that  they  are  symmetric,  that  is, 
that  one  side  is  exactly  like  the  other. 

In  this  case  we  say  that  the  figure  is  symmetric  with 
respect  to  an  axis,  this  axis  being  the  crease  in  the  paper 
or,  more  generally,  the  line 
which  divides  the  figure  into 
two  parts  that  will  fit  each 
other  if  folded  over. 

In  architecture  we  often  find 
symmetry  with  respect  to  an 
axis.  For  example,  in  this 
picture  of  the  interior  of  a 
great  cathedral  we  see  that 
much  of  the  beauty  and  gran- 
deur is  due  to  symmetry. 

This  case  is  evidently  one 
of  symmetry  with  respect  to  a 
plane  instead  of  with  respect 
to  a  line.  We  may  also  have  symmetry  with  respect  to  a 
center,  that  is,  a  figure  may  turn  halfway  round  a  point 
and  appear  exactly  as  at  first.  This  is  seen  in  a  circle,  or, 
among  solids,  in  a  sphere.  It  is  also  seen  in  the  Gothic 
window  shown  on  page  148.  Symmetry  of  all  kinds  plays 
a  very  important  .part  in  art,  not  merely  in  architecture, 
painting,  and  sculpture,  but  in  all  kinds  of  decoration. 


148 


GEOMETKY  OF  FOKM 


Exercise  16.   Symmetry 

1.  Has  this  Gothic  window  an  axis  of  symmetry?    If  so, 
draw  the  circle  and  indicate  the  axis  of  symmetry.    If  it 
has  more  than  one  axis  of  symmetry, 

draw  each  axis  of  symmetry. 

2.  If  the  figure  has  a  center  of 
symmetry,    indicate    this   center    in 
your  rough  sketch  by  the  letter  0. 

3.  Draw   an   equilateral   triangle 
and  draw  all  its  axes  of  symmetry. 

4.  Draw  a  square  and  draw  all 
its  axes  of  symmetry. 

5.  Draw  a  plane  figure  with  no  axis  of  symmetry ;  one 
having  only  one  axis  of  symmetry ;  one  having  two  axes 
of  symmetry;  one  having  any  number  of  axes  of  symmetry. 

6.  Draw  the  following  designs  in  outline  and  indicate 
by  letters  all  the  axes  of  symmetry  in  each  design: 


7.  Write  a  list  of  three  windows  in  churches  in  your 
locality  which  have  axes  of  symmetry.  If  you  know  of 
any  window  which  has  a  center  of  symmetry,  mention  it. 

The  class  should  be  asked  to  mention  other  illustrations  of  axes 
of  symmetry,  as  in  doors  and  in  linoleum  patterns.  There  should 
also  be  questions  concerning  planes  of  symmetry,  as  in  a  cube,  a 
sphere,  a  chair,  animals,  and  vases.  Objects  in  the  schoolroom 
offer  a  good  field  for  inquiry. 


CUKVES 


149 


Plane  Figures  formed  by  Curves.  We  have  already  men- 
tioned a  number  of  figures  formed  by  curve  lines  without 
attempting  to  define  them.  We  shall  now  mention  these 
again  and  shall  discuss  more  fully  a  few  of  those  which 
occur  most  frequently  in  drawing,  pattern 
making,  architecture,  measuring,  and  the  like. 

This  figure  represents  a  circle  with  center 
0,  radius  OA,  and  diameter  BC. 

The  circle  is  sometimes  thought  of  as  the 
space  inclosed  and  sometimes  as  the  curve 
line  inclosing  the  space.    The  length  of  this 
curve  is  called  the  circumference,  and  sometimes  the  curve 
itself  is  called  by  this  name. 

It  is  not  expected  that  the  above  statement  will  be  considered  as 
a  formal  definition  to  be  learned.  All  that  is  needed  at  this  time  is 
that  the  terms  shall  be  used  properly.  Teachers  should  recognize 
that  circle  and  circumference  both  have  two  meanings,  as  stated  above. 

Another  interesting  figure,  but  one  which  is  used  not 
nearly  so  often  as  the  circle,  is  the  ellipse.  If  we  place  two 
thumb  tacks  at  A  and  B,  say  3  in. 
apart,  and  fasten  to  them  the  ends 
of  a  string  which  is  more  than 
3  in.  long,  draw  the  string  taut 
with  a  pencil  point  P,  and  then 
draw  the  pencil  round  while  keep- 
ing the  string  taut,  we  shall  trace 
the  ellipse. 

It  is  evident  that  an  ellipse  has  two  axes  of  symmetry 
and  one  center  of  symmetry. 

The  orbits  of  the  planets  about  the  sun  are  ellipses. 

When  facilities  for  drawing  permit,  the  student  should  draw  ellip- 
ses of  various  sizes  and  shapes  and  should  satisfy  himself  that  two 
ellipses  are  not  in  general  similar. 


150 


GEOMETRY  OF  FOKM 


Solids  bounded  by  Curved  Surfaces.  We  have  often 
mentioned  the  sphere,  and  shall  now  speak  of  it  and  of 
other  solids  bounded  in  whole  or 
in  part  by  curved  surfaces. 

A  sphere  is  a  solid  bounded  by 
a  surface  whose  every  point  is 
equidistant  from  a  point  within, 
called  the  center. 

We  also  speak   of  the  radius 

and  diameter  of  a  sphere,  just  as  we  speak  of  the  radius 
and  diameter  of  a  circle. 

A  cylinder  is  a  solid  bounded  by  two 
equal  circles  and  a  curved  surface  as 
shown  in  this  figure. 

The  two  circles  which  form  the  ends 
are  called  the  bases  of  the  cylinder  and 
the  radius  and  diameter  of  either  base 
are  called  respectively  the  radius  and 
diameter  of  the  cylinder. 

The  line  joining  the  centers  of  the  two  bases  is  called 
the  axis  of  the  cylinder,  and  its  length 
is   called  the   height   or    altitude  of   the 
cylinder. 

A  cone  is  a  solid  like  the  one  here 
shown.  It  has  a  circular  base  and  an 
axis  of  symmetry  from  the  center  of  the 
base  to  the  vertex  of  the  cone.  The  per- 
pendicular distance  from  the  vertex  to 
the  base  is  called  the  height  or  altitude 
of  the  cone.  The  words  radius  and 
diameter  are  used  as  with  the  cylinder. 

In   this   book  we    shall   consider   only  cylinders    and   cones  in 
which  the  axes  are  perpendicular  to  the  bases. 


SOLIDS  BOUNDED  BY  CURVES  151 

Exercise  17.   Solids  bounded  by  Curves 

1.  If  you  cut  off  a  portion  of  a  sphere,  say  a  wooden 
ball,  by  sawing  directly  through  it,  but  not  necessarily 
through  the  center,  what  is  the  shape  of  the  flat  section  ? 
Illustrate  by  a  drawing. 

2.  In  Ex.  1  is. the  section  always  a  plane  of  symmetry? 
If  not,  is  it  ever  a  plane  of  symmetry,  and  if  so,  when  ? 

3.  A  cylinder  is  symmetric  with  respect  to  what  line 
or  lines  ?    with  respect  to  what  plane  or  planes  ? 

4.  Could  a  cylindric  piece  of  wood,  say  a  broom  handle, 
be  so  cut  that  the  section  would  be  a  circle  ?   If  so,  how 
should  it  be  cut  ?   Could  it  be  so  cut  that  the  section  would 
seem  to  be  an  ellipse  ?    Illustrate  each  answer. 

The  section  last  mentioned  is  really  an  ellipse,  and  this  is  proved 
in  higher  mathematics. 

5.  Could  a  cylindric  piece  of  wood  be  so  cut  that  the 
section  would  be  a  rectangle  ?   a  trapezoid  ?   Illustrate. 

6.  Three  cylinders  of  the   same  height,  4  in.,  have  as 
diameters  3  in.,  4  in.,  and  5  in.  respectively.    Can  a  section 
in  any  one  of  them  be  a  square  ?    Illustrate  the  answer. 

7.  Is  a  cone  symmetric  with  respect  to  any  line  ?   to 
any  plane  ?   Illustrate  each  answer. 

8.  How  could  a  cone  be  cut  so  as  to  have  the  section 
a  circle  ?    a  triangle  ?   apparently  an  ellipse  ?    Illustrate. 

The  section  last  mentioned  is  really  an  ellipse,  and  this  is  proved 
in  higher  mathematics.  This  is  the  reason  why  an  ellipse  is  called 
a  conic  section.  Other  conic  sections  are  studied  in  higher  mathe- 
matics, and  they  are  important  in  the  study  of  astronomy,  mechanics, 
and  other  sciences. 

9.  How  is  the  largest  triangle   obtained  by  cutting  a 
cone  ?    Illustrate  the  answer. 


152 


GEOMETRY  OF  FORM 


Exercise  18.    Review 

1.  If    the   rays   of    light    from   any   object   ABC  pass 
through  a  small  aperture  0  of  an  opaque  screen  and  fall 
upon  another  screen  parallel  to  the 

object,  an  inverted  image  A'B'C'   c 

will  be  formed  as  here  shown.    If  B 

the  object  is  5  ft.  long  and  9  ft.  A 

from  0,  how  far  from  0  must  the 

second  screen  be  placed  so  that  the  image  shall  be  6  in. 

long  ?    How  far,  so  that  the  image  shall  be  8  in.  long  ? 

2.  With  the  aid  of  ruler  and  compasses,  construct  figures 
similar  to  each  of  the  following  figures,  but  twice  as  large, 
and  indicate  the  axis,  axes,  or  center  of  symmetry  of  each. 


3.  Draw  the  following  figure  about  half  as  large  again 
and  make  it  the  basis  for  a  pattern  for  lino- 
leum, using  other  lines  as  necessary. 

4.  Draw  a  square  and  cut  it  into  four  tri- 
angles by  means  of  two  diagonals.    Describe 
the  triangles  with  respect  to  their  being  equi- 
lateral, isosceles,  or  right. 

5.  Draw  this  figure  about  half  ,as  large         r      \ 
again  and  make  it  the  basis  for  a  pattern        J^      J-^ 
for  a  church  window,  using  other  lines  as  (  j 
may  be  necessary  for  the  purpose.  ^_^^_x 


OUTDOOR  PROBLEMS  153 

Exercise  19.    Optional  Outdoor  Work 

1.  Collect,   if  possible,  several   leaves  of  each  of  the 
following  kinds  of  tree :  oak,  elm,  maple,  pine,  and  poplar. 
Are  the  leaves  of  each  kind  of  tree  approximately  similar 
to  the  other  leaves  of  the  same  tree  ?    Has  each  of  the 
leaves  an  axis  of  symmetry  ? 

2.  Do  you  know  of  any  building  lots  or  fields  that  are 
triangular?    If  so,  make  rough  outline  drawings  of  them. 

3.  Do  any  of  the  public  buildings  of  your  community 
have  cylindric  columns  ?    If  so,  which  buildings  ? 

4.  Do  you  know  of  any  church  spires  that  are  conic  in 
shape  ?    If  so,  which  spires  ? 

5.  What  is  the  shape  most  frequently  used  in  decorat- 
ing the  interiors  of  churches  in  your  vicinity  ? 

6.  Can  you  find   an  illustration  of  a  Gothic  window 
in  any  of  the  churches  in  your  vicinity  ?    If  so,  where  ? 

7.  If  there  is  a  standpipe  in  your  vicinity,  what  is  its 
shape  ?    What  is  the  shape  of  most  of  the  smokestacks  of 
the  factories  in  your  community  ? 

8.  Notice  the  designs  in  the  carpeting,  wall  paper,  and 
linoleum  exhibited  by  various  stores.   What  general  pattern 
or  design  is  most  frequently  used  ? 

9.  If  convenient,  inspect  a  house  that  is  being  built 
and  compare  the  floor  plan  with  the  plans  of  the  contractor 
or  architect.   What  scale  was  used  in  drawing  the  plan? 

10.  Name  illustrations  of  each  of  the  following  forms 
that  you  have  seen  in  your  community:  circle,  rectangle, 
cylinder,  cone,  sphere,  trapezoid,  and  triangle. 

As  stated  above,  this  work  is  purely  optional.  It  is  suggestive  of 
a  valuable  line  of  local  questions. 


154  GEOMETRY  OF  FORM 

Exercise  20.    Problems  without  Figures 

1.  How  do  you  construct  a  triangle,  having  given  the 
lengths  of  the  three  sides  ? 

2.  How  do  you  construct  an  isosceles  triangle,  having 
given  the  base  and  one  of  the  equal  sides  ? 

3.  How  do  you  construct  an  equilateral  triangle,  having 
given  one  of  the  sides  ? 

4.  State  two  methods  of  drawing  from  a  given  point  a 
perpendicular  to  a  given  line. 

5.  If  you  have  a  line  drawn  on  paper,  what  is  the  best 
way  you  know  to  bisect  it  ? 

6.  How  do  you  bisect  an  angle  ? 

7.  How  do  you  construct  an  angle  exactly  equal  to  a 
given  angle  ? 

8.  How  do  you  draw  a  line  parallel  to  a  given  line  ? 

9.  If  you   have  a  line  drawn   on  paper  and  wish  to 
divide  it  into  five  equal  parts,  how  do  you  proceed  ? 

10.  How  do  you  construct  a  six-sided  figure  in  a  circle, 
the  sides  all  being  equal  ? 

11.  How  do  you  draw  to  a  given  scale  the  rectangular 
outline  of  the  printed  part  of  this  page  ? 

12.  How  do  you  draw  a  plan  of  the  top  of  your  desk  to 
the  scale  of  a  certain  number  of  inches  to  a  foot  ? 

13.  When  you  know  the  scale  which  was  used  in  draw- 
ing a  map,  how  do  you  find  the  actual  distance  between 
two  cities  which  are  shown  on  the  map  ? 

14.  How  can  you  enlarge  a  drawing   by  the  aid  of  a 
pantograph  ? 

15.  How  do  you  determine  whether  a  figure  is  symmetric 
with  respect  to  an  axis  ? 


GEOMETRY  OF  SIZE  155 

II.   GEOMETRY  OF  SIZE 

Size.  On  page  111  we  found  that  geometry  is  concerned 
with  three  questions  about  any  object :  What  is  its  shape  ? 
How  large  is  it  ?  Where  is  it  ?  Thus  far  we  have  con- 
sidered the  shape  of  objects;  we  shall  now  consider  size. 

There  are  several  ideas  to  be  considered  when  we  think 
and  speak  of  the  size  of  objects,  such  as  length,  area,  and 
volume,  all  of  which  we  may  include  in  the  single  expres- 
sion geometric  measurement.  That  is,  we  shall  not  think 
of  size  as  including  the  measurement  of  weight,  of  value, 
of  hardness,  and  the  like,  but  only  as  including  the  length 
(width,  height,  depth,  and  distance  in  general),  area 
(surface),  and  volume  (capacity)  of  figures. 

Length.  It  seems  very  easy  to  measure  accurately  the 
length  of  anything,  but  it  is  not  so  easy  as  it  seems. 
Linen  tape  lines  stretch,  steel  tape  lines  contract  in  cold 
weather,  ordinary  wooden  rulers  shrink  a  little  when  they 
get  very  dry,  and  chains  wear  at  the  links  and  thus 
become  longer  with  age.  But  these  matters  are  of  less 
moment  than  the  carelessness  of  those  who  make  the 
measurements.  If  the  members  of  your  class,  each  by 
himself,  should  measure  the  length  of  the  walk  in  front 
of  your  school,  to  the  nearest  sixteenth  of  an  inch,  and 
not  compare  results  until  they  had  finished,  it  is  likely 
that  each  would  have  a  different  result.  In  fact,  all 
measurement  is  simply  a  close  approximation. 

One  of  the  best  ways  of  securing  a  close  approximation 
to  the  true  result  is  to  make  the  measurement  in  two 
different  ways.  Never  fail  to  check  a  measurement. 

Just  as  we  should  always  check  an  addition  by  adding  in  the 
opposite  direction,  so  we  should  always  check  a  measurement  of 
length  by  measuring,  if  possible,  in  the  opposite  direction. 


156  GEOMETRY  OF  SIZE 

Outdoor  Work.  In  connection  with  the  study  of  the 
size  and  position  of  common  forms  we  shall  first  suggest 
a  certain  amount  of  work  to  be  done  out  of  doors. 

1.  Measure  the  length  of  the  school  grounds. 

To  do  this,  drive  two  stakes  at  the  appropriate  corners, 
putting  a  cross  on  top  of  each  stake  so  as  to  get  two  points 
between  which  to  measure. 

Measure  from  A  to  B  by     ^^ 

holding  the  tape  taut  and  ^^^^    '~\__  n 

level,  drawing  perpendic- 
ulars when  necessary  by  means  of  a  plumb  line  as  shown 
in  the  figure.    Check  the  work  by  measuring  from  B  back 
to  A  in  the  same  way. 

2.  Run  a  straight  line  along  the  sidewalk  in  front  of 
the  school  yard. 

Of  course  for  a  short  distance  this  is  easily  done  by 
stretching  a  string  or  a  measuring  tape,  but  for  longer 
distances  another  plan  is 
necessary.  x~       ~p  Q          R  Y 

If  we  wish  to  run  a  line 

from  X  to  F,  say  300  ft.,  we  drive  stakes  at  these  points 
and  mark  a  cross  on  the  top  of  each  so  as  to  have  exact 
points  from  which  to  work.  Now  have  one  student  stand 
at  X  and  another  at  Y",  each  with  a  plumb  line  marking  the 
exact  points.  Then  have  a  third  student  hold  a  plumb  line 
at  some  point  P,  the  student  at  X  motioning  him  to  move 
his  plumb  line  to  the  right  or  to  the  left  until  it  is  exactly 
in  line  with  X  and  Y.  A  stake  is  then  driven  at  P,  and 
the  student  at  X  moves  on  to  the  point  P.  The  point  Q 
is  then  located  in  the  same  way.  In  this  manner  we  stake 
out  or  "  range  "  the  line  from  X  to  Y,  checking  the  work 
by  ranging  back  from  Y  to  X. 


OUTDOOR  WORK  157 

3.  Measure  the  height  of  a  tree  by  making  on  the  ground 
a  right  triangle  congruent  to  a  right  triangle  which  has  the 
tree  as  one  side. 

To  do  this,  sight  along  an  upright  piece  of  cardboard 
so  as  to  get  the  angle  from  the  ground  to  the  top  of  the 
tree.  Mark  tne  angle  on  the  cardboard  and  then  turn  the 
cardboard  down  flat  so  as  to  have  an  equal  angle  on 
the  ground.  A  right  triangle  can  now  easily  be  laid  out 
on  the  ground  so  as  to  be  congruent  to  the  one  of  which 
the  tree  is  one  side.  By  measuring  a  certain  side  of  this 
right  triangle,  the  height  of  the  tree  can  be  found. 

4.  Run  a  line  through  a  point  P  parallel  to  a  given 
line  AB  for  the  purpose  of  laying  out  one  of  the  two  sides 
of  a  tennis  court. 

P  O 

Stretch  a  tape  line     

from  P  to  any  point 
M  on  AB,  bisect  the 
line  PM  at  0,  and 

from  any  point  N  on     -^ N  ,,  —   — ^ 

AB  draw  NO.   Pro- 
long NO  to  Q,  making  OQ  equal  to  NO,  and  draw  PQ.  Sup- 
pose that  ON  is    20  ft.    Then   sight  from  N  through  0, 
and  place  a  stake  at  Q  just  20  ft.  from  0.    Then  P  and 
Q  determine  a  line  parallel  to  AB. 

The  proof  of  this  fact,  like  the  proofs  of  many  other  facts  inferred 
from  certain  of  the  exercises,  is  part  of  demonstrative  geometry, 
which  the  student  will  meet  later  in  his  course  in  the  high  school^ 

Outdoor  work  will  be  given  at  intervals  and  always  by  itself,  so 
that  it  can  easily  be  omitted.  The  circumstances  vary  so  much  in  dif- 
ferent parts  of  the  country  as  to  climate,  location  of  the  school,  and 
other  conditions,  that  a  textbook  can  merely  suggest  work  of  this 
kind  which  may  or  may  not  be  done,  as  the  teacher  directs.  A  good 
tape  line,  three  plumb  lines  (lines  with  a  piece  of  lead  at  one  end), 
and  a  pole  abovit  10  ft.  long  will  serve  for  an  equipment  for  beginners. 


158  GEOMETRY  OF  SIZE 

Exercise  21.   Practical  Measurements  of  Length 

1.  Measure  the  length  of  this  page  to  the  nearest  thirty- 
second  of  an  inch,  checking  the  work. 

If  a  ruler  is  used  which  is,  as  usual,  divided  only  to  eighths  of 
an  inch,  the  student  will  have  to  use  his  judgment  as  to  the  nearest 
thirty-second  of  an  inch.  The  protractor  illustrated  on  page  115  has 
an  edge  on  which  lengths  are  given  to  sixteenths  of  an  inch,  and 
such  a  scale  may  be  used  if  laid  along  the  edge  of  a  ruler. 

2.  Measure  the  length  of  this  page  to  the  nearest  twen- 
tieth of  an  inch,  checking  the  work. 

The  protractor  illustrated  on  page  115  has  an  edge  divided  into 
tenths  of  an  inch.  There  is  advantage  in  the  student  becoming 
familiar  with  the  units  of  the  metric  system,  even  before  he  studies 
the  subject  on  page  205,  since  these  units  have  come  into  use  in 
our  foreign  trade  and  in  all  our  school  laboratories.  It  is  desirable 
to  know  that  10  millimeters  (mm.)  =  1  centimeter  (cm.)  =  0.4  in., 
nearly;  10  cm.  =  1  decimeter  (dm.);  10  dm.  =  1  meter (m.)  =  39.37 in. 

3.  Measure  the  length  of  the  longest  line  of  print  on 
this  page,  to  the  nearest  sixteenth  of  an  inch,  checking 
the  work. 

If  the  student  has  a  pair  of  dividers  (compasses  with  sharp  points), 
this  may  be  used  to  transfer  the  length  to  a  ruler.  This  method 
is  usually  more  nearly  accurate  than  to  lay  the  ruler  on  the  page. 

4.  Measure  the  length  of  your  schoolroom  to  the  nearest 
eighth  of  an  inch,  checking  the  work. 

If  the  class  works  in  groups  of  two,  and  each  group  checks  its 
result  with  care,  there  may  still  be  some  difference.  In  that  case  an 
average  may  be  taken.  This  is  commonly  done  in  surveying. 

5.  In  the  upper  part  of  the  opposite  picture  a  man  is 
sighting  across  the  stream  in  line  with  the  front  part  of 
his  cap.    He  then  turns  and  sights  along  the  ground,  as 
shown  by  the  other  man  standing  near  him.    How  does  he 
find  the  width  of  the  stream  by  this  method  ? 


LENGTH 


159 


(Curious  illustrations  from  old  geometries 

if-  the  XVI  century.  The  first  one  shows  howto  measure  the  distance  across 

a  stream.  <A  soldier  is  said  to  haTie  helped  Napoleon  in  this  -way 

in  one  of  his  military  campaigns.  The  second  one  shows  how 

to  measure  the  height  of  a  tower  with  the  aid  of  a  drum. 


160  GEOMETRY  OF  SIZE 

6.  Draw   a   line    6  in.   long,    and   on   it   measure   off 
2.8  in.  from  one  end  and  2.3  in.  from  the  other  end.   Check 
the  results  by  measuring  the  length  of  the  intermediate 
portion.    What  should  that  length  be  ?    What  do  you  find 
it  to  be  by  actual  measurement  ? 

7.  Draw  a  line   3.9  in.  long,   and  on  it  measure  off 
successive  lengths  of  1^-  in.  and  1-|  in.    Check  the  results 
by  measuring  the  length  of  the  remaining  portion,  as  in 
Ex.  6.    What  should  that  length  be  ?    What  do  you  find 
it  to  be  by  actual  measurement? 

8.  Draw  a  line  4y^-  in.  long,  and  on  it  measure  off  a 
line    2-^2"  in.   long.     Check   the   results   by  bisecting   the 
original  line  as  on  page  124  and  seeing  if  the  point  of 
bisection  falls  at  the  end  of  the  part  measured  off. 

9.  Draw   a   line   9^  in.  long,   and  on  it   measure    off 
successive  equal  lengths  of  3^  in.    Check  the  results  by 
dividing  the  original  line  into  three  equal  parts  by  the 
first  method  given  on  page  130. 

10.  Construct  a  square  1  in.  on  a  side  by  the  methods 
already  learned,  and  measure  the  length  of  each  diagonal. 
What  do  you  find  it  to  be  by  actual  measurement  ?    If  the 
square  is  accurately  constructed,  what  must  be  the  length 
of  each  diagonal  to  the  nearest  tenth  of  an  inch;  that  is, 
what  is  the  square  root  of  2? 

11.  Construct   a   rectangle   3  in.  high    and   4  in.  long. 
Check  the  accuracy  of  the  construction  by  measuring  the 
length  of  each  diagonal.    What  do  you  find  it  to  be  by 
actual  measurement  ? 

Such  measurements  give  some  idea  of  the  accuracy  required  in  a 
machine  shop.  With  the  instruments  which  the  students  have,  these 
measurements  are  as  nearly  accurate  as  can  be  required,  but  for 
practical  purposes  much  closer  approximations  are  often  necessary. 


LENGTH 


161 


ea  more  than  a 
Id  saying, "  Give  him 


162 


GEOMETKY  OF  SIZE 


Estimates  of  Area.  There  are  several  methods  for  esti- 
mating areas,  of  which  we  shall  now  consider  two.  In 
general  it  will  be  found  that  neither  of  these  plans  is 
very  practical,  although  both  are  used  in  certain  difficult 
cases  of  measurement.  The  most  practical  way  of  finding 
areas  is  introduced  on  page  164. 

• 

1.  If  an  area  inclosed  by  a  curve  is  drawn  on  squared 
paper,  the  area  may  be  estimated  approximately  by  count- 
ing the  squares  contained  within  the  curve.    The  squares 
on  the  boundary  should  be  included  or  excluded  according 
as  more  than  half  their  area  is  included  or  is  not  included 
within  the  bounding  line,  and  half  the  other  squares  that 
are  practically  half  within    and  half  without  should  be 
included.    In  certain  cases  the  rule  should  be  altered  as 
the  peculiarity  of  the  case  requires. 

For  example,  in  this  figure  if  each  square  represents  1  sq.  in.,  the 
area  inclosed  by  this  curve  is  approximately  33  sq.  in.,  there  heiv 
approximately  33  squares  inclosed.  

Paper  such  as  that  used  in  the  illustration 
is  called  squared  paper  or  cross-section  paper.  It 
can  generally  be  bought  at  any  stationer's. 
When  the  paper  is  ruled  into  squares  one  tenth 
of  an  inch  on  each  side,  there  are,  of  course, 
100  such  squares  in  1  sq.  in. 

In  case  it  is  not  easy  to  purchase  squared 
paper,  ruled  in  tenths  of  an  inch,  it  is  advis- 
able for  the  student  to  rule  some  paper,  drawing  the  lines  with 
the  same  care  taken  in  making  the  other  constructions  of  geometry. 

2.  The  area  inclosed  by  a  curve  drawn  on  thick  paper  or 
cardboard  may  be  estimated  by  cutting  out  the  area  to  be 
measured,  weighing  it,  and  comparing  its  weight  with  that 
of  a  unit  of  area,  such  as  a  square  inch  cut  from  the  paper. 

Since  this  method  considers  weight,  it  is  not  geometric,  and 
furthermore  it  is  not  very  practical  in  'estimating  small  areas. 


ESTIMATES  OF  AREAS  163 

Exercise  22.    Estimates  of  Areas 
1.  Estimate  the  area  of  each  of  the  following  figures: 


2.  Draw  the  outline  of  a  leaf  on  squared  paper  and 
estimate  the  area. 

Paper  often  comes  ruled  in  millimeters,  in  which  case  the  areas 
can  be  found  with  greater  accuracy,  to  square  millimeters. 

3.  On  a  piece  of  squared  paper  draw  a  rectangle  2  in. 
long  and  1.7  in.  wide,  and  divide  it  into  two  triangles  by 
drawing  either  diagonal.    Estimate  the  area  of  each  triangle, 
state  whether  the  areas  are  equal,  and  check  the  work  by 
finding  the  area  of  the  rectangle  and  showing  that  this  is 
equal  to  the  sum  of  the  areas  of  the  triangles. 

4.  On  a  piece  of  squared  paper  draw  a  parallelogram 
1.9  in.  long  and  1  in.  high.    Draw  either  diagonal,  estimate 
the  area  of  each  triangle  thus  formed,  and  proceed  further, 
as  in  Ex.  3. 

5.  On  a  piece  of  squared  paper  draw  a  trapezoid  1  in. 
high,  with  lower  base  2  in.  and  upper  base  1.2  in.     Esti- 
mate the  area  of  the  trapezoid  by  the  method  of  Ex.  3. 

This  method  of  approximation  for  estimating  areas  is  sufficient 
for  many  purposes,  but  the  methods  of  geometry,  some  of  which 
we  shall  now  study,  are  greatly^superior  to  this  method. 


164 


GEOMETRY  OF  SIZE 


1  square  inch 
1  sq.  in. 


lin. 


Unit  of  Area.    We  have  seen  how  we  may  estimate  an 
area  to  a  fair  degree  of  accuracy.    Whether  we  estimate 
or  actually  measure,  we  commonly  ex. 
press  the  area  by  means  of  some  unit 
square,  such  as  the  square  inch. 

A  square  inch  is  not  the  same  as  1  in. 
square ;  1  sq.  in.  is  the  area  of  a  space  that 
is  1  in.  square.  A  circle  may  have  this  area. 

There  are  also  other  units  of  area, 
such  as  the  acre  (160  sq.  rd.). 

Area  of  a  Rectangle.  A  school  building  has  a  rectangular 
entrance  hall  10  ft.  long  and  4  ft.  wide,  the  floor  being  made 
of  marble  squares  1  ft.  on  a 
side.    What   is   the   easiest 
way  of  finding  the  area  of 
the  floor  ? 

There  are  10  squares  in 
each  row  and  there  are  4 
rows  of  squares.  Since  there  are  4  x  10  squares,  we  have: 

Area  =  4  x  10  sq.  ft.  =  40  sq.  ft. 

The  area  of  a  rectangle  is  equal  to  the  product  of  the  base 
and  height. 

This  means  that  10  x  4  —  40,  the  number  of  square  feet. 

We  often  express  this  statement  by  &  formula,  using  A 
for  area,  b  for  base,  and  h  for  height,  thus: 

A  =  b  x  h, 
or,  briefly,  A  =  bh. 

The  absence  of  a  sign  between  letters  in  a  formula  indicates 
multiplication. 

In  this  book  rooms,  boxes,  fields,  and  the  like  are  to  be  considered 
as  rectangular  unless  the  contrary  is  stated. 


AREA  OF  A  RECTANGLE  165 

Exercise  23.   Area  of  a  Rectangle 

Using  the  formula  given  on  page  164,  find  the  areas  of  the 
following  rectangles  : 

1.  18£  ft.  by  26  ft.  5.  36.3  ft.  by  142.5  ft. 

2.  27  yd.  by  42  J  yd.  6.  12.8  ft.  by  17.3  ft. 

3.  13Jrd.  by  28  rd.  7.  42.3  yd.  by  46.8  yd. 

4.  1\  in.  by  9|  in.  8.  37J  in.  by  62  in. 

y  9.  Find  the  floor  area  of  a  room  that  is  28  ft.  6  in.  long 
and  32  ft.  wide. 

10.  Find  the  area  of  a  sheet  of  paper  3^  in.  square. 
Verify  the  result  by   drawing   the   figure   and  ruling  it 
off  into  ^-inch  squares. 

11.  A  garden,  38  ft.  by  56  ft.,  contains  a  3-foot  walk 
laid  inside  the  garden  along  the  four  sides.   The  mid-points 
of  the  long  sides  are  joined  by  a  2-foot  path.     Find  the 
area  left  for  cultivation,  and  draw  a  plan  to  scale. 

12.  On  the  floor  of  a  room  32ft.  long  and  24ft.  wide 
a  border  2  ft.  wide  is  to  be  painted.     Find  the   cost  of 
painting  the  border  at  300  per  square  yard. 

^13.  At  160  per  square  foot  find  the  cost  of  cementing 
a  walk  6  ft.  wide  round  the  outside  of  a  garden  that 
measures  56  ft.  by  82  ft. 

^  14.  A  student,  being  asked  to  measure  a  rectangle,  under- 
stated the  length  2%  and  overstated  the  width  3%.  Find 
the  per  cent  of  error  in  the  area  computed. 

15.  Draw  a  plan  of  the  floor  of  the  basement  of  a  house 
to  scale  as  follows :  Start  at  A,  go  north  14  ft.  to  B,  west 
8  ft.  to  (7,  north  5  ft.  to  Z>,  west  8  ft.  to  E,  south  19  ft. 
to  F,  and  east  to  A.  Find  the  cost  of  cementing  the  floor 
at  150  per  square  foot. 


166  GEOMETRY  OF  SIZE 

16.  Construct  a  rectangle  and  then  construct  a  similar 
rectangle  the  area  of  which  is  three  times  that  of  the  first. 

17.  The  screens  A,  B,  and  C  are  1  ft.,  2  ft.,  and  3  ft. 
respectively  from  an  electric  light  L.    If  screen  A  should  be 
removed,  the  quantity  of  light  which  fell 

on  it  would  fall  on  B.  If  screens  A  and 
B  should  be  removed,  the  same  quantity 
of  light  would  fall  on  screen  C.  How 
would  the  intensity  of  light  compare  for 
a  given  area  on  each  of  the  screens  ? 

18.  How  many  paving  blocks   each  4  in.  by  4  in.  by 
10  in.,  placed  on  their  sides,  will  be  required  to  pave  a 
street  1800  ft.  long  and  34  ft.  8  in.  wide  ? 

19.  Printers  usually  cut  business  cards  from  sheets  22  in. 
by  28  in.    How  many  cards  2  in.  by  4  in.  can  be  cut  from 
one  of  these  sheets  ?    Draw  a  plan. 

20.  At  6$  a  square  foot  find  the  cost  of  enough  wire 
screen  for  the  8  windows  of  a  gymnasium,  each  being  28  in. 
by  64  in.  inside  the  frame.    Allowance  should  be  made  for 
the  wire  to  overlap  the  frames  1  in.  on  every  side. 

21.  Find  the  area  of  a  double  tennis  court. 
A  standard  double  tennis  court  is  36  ft.  by  78  ft. 

22.  What  is  the  meaning  of  the  statement  A=bh? 

23.  How  many  acres  are  there  in  a  football  field  ? 

A  standard  football  field  is  100  yd.  by  53  yd.  1  ft. 
An  acre  is  160  sq.  rd.,  and  a  rod  is  5^  yd.   The  teacher  should  give 
plenty  of  practical  work  in  finding  the  areas  of  floors  and  the  like. 

24.  A  farm  team  plowing  a  field  walks  at  the  rate  of 
2  mi.  per  hour  and  is  actually  plowing  -|  of  the  time.  What 
area  will  be  plowed  from  7  A.M.  to  noon  if  a  plow  is  used 
which  turns  a  furrow  14  in.  wide  ? 


AREA  OF  A  PARALLELOGRAM 


167 


Area  of  a  Parallelogram.  It  is  often  convenient  or  nec- 
essary to  find  the  area  of  a  parallelogram. 

If  from  any  parallelogram,  like  ABCD  in  the  first  figure, 
\ve  cut  off  the  shaded  triangle  T  by  a  line  perpendicular 


D 


to  DC,  and  place  the  triangle  at  the  other  end  of  the  paral- 
lelogram, as  shown  in  the  figure  at  the  right,  the  resulting 
figure  is  a  rectangle. 

That  is,  the  area  of  a  parallelogram,  is  equal  to  the  area 
of  a  rectangle  of  the  same  base  and  the  same  height.  But 
the  formula  for  the  area  of  a  rectangle  is  A  =  bh. 

Therefore  the  area  of  a  parallelogram  is  equal  to  the 
product  of  the  base  and  height. 

This  may  be  expressed  by  the  formula 
A  =  bh. 

The  teacher  should  make  sure  that  the  students  understand  the 
meaning  of  this  formula.  The  purpose  is  to  introduce  algebraic 
forms  as  needed.  The  students  should  see  that  the  value  of  A 
depends  upon  the  values  of  b  and  h.  In  the  language  of  more 
advanced  mathematics,  A  is  called  a  function  of  b  and  h.  The 
students  should  see  that  all  formulas  are  expressions  of  functions. 


1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

[ 

1 

J 

1 

J 

1 

Rectangular  pieces  of  cardboard,  as  in  the  figures  shown  just 
above,  may  be  arranged  to  lead  the  student  to  infer  that  when 
the  base  and  height  of  a  rectangle  are  equal  respectively  to  the 
base  and  height  of  a  parallelogram,  the  areas  are  equal. 


168  GEOMETRY  OF  SIZE 

Exercise  24.  Area  of  a  Parallelogram 

1.  Draw  parallelograms  of  the  shapes  and  sizes  of  the 
following  and  show,  by  cutting  off  triangles  and  placing 
them  as  explained  on  page  167,  that  each  parallelogram 
can  be  transformed  into  a  rectangle  of  the  same  area. 


2.  On  squared  paper  draw  four  parallelograms  and  a 
rectangle,  all  having  equal  bases  and  equal  heights,  but 
all  of  different   shapes.     By  cutting  off    a  triangle  from 
each  parallelogram  and  moving  it  to  the  other  side,  trans- 
form each  into  a  rectangle  of  the  same  area.    Count  the 
squares  and  compare  the  areas  of  the  resulting  rectangles. 

Draw  the  following  parallelograms  to  scale  and  find  the 
area  of  each: 

3.  Base  30  in.;  other  side,  20  in.;  height,  15 in.;  scale  ^-. 

4.  Base  12  in. ;  other  side,  8  in. ;  height,  6  in. ;  scale  4-. 

5.  On  squared  paper  draw  a  rectangle  and  a  parallelo- 
gram with  equal  bases  and  equal  heights.    Compute  the 
area  of  each  by  counting  the  included  squares,  and  thus 
compare  the  areas. 

6.  A  floor  is  paved  with  six-sided 
tiles,  as  here  shown.    The  tiles  have 
been  divided  by  dotted  lines  in  the 
picture   to    suggest    a    method    of 

measuring  them.  What  measurements  would  you  take  to 
find  the  area  of  each  tile  ?  What  other  divisions  of  the 
tiles  can  you  suggest  for  convenience  in  finding  the  area 
of  each? 


AEEA  OF  A  TRIANGLE  169 

Exercise  25.   Area  of  a  Triangle 

1.  How  is  the  area  of  a  parallelogram  found? 

2.  In  the  parallelogram  here  shown  how  do  the  areas 
of  the  triangles  ABC  and  CD  A  compare  ?    A  triangle  is 
what  part  of  a  parallelogram  of  the  same 

base  and  height  ?  / ^7 

/  9***         / 

The  parallelogram  should  be  cut  out  of  paper       /       ,,---''          I 

and  then  divided  into  two  congruent  triangles      1^- J 

by  cutting  along  one  of  the  diagonals. 

3.  If  the  parallelogram  in  Ex.  2  is  6  in.  wide  and  3  in. 
high,  what  is  its  area?    What  is  the  area  of  each  of  the 
triangles  formed  by  drawing  the  diagonal  AC? 

4.  If  a  parallelogram  is  5  ft.  wide  and  2  ft.  high,  what 
is  its  area  ?    What  is  the  area  of  each  of  the  triangles  ? 

5.  If  a  parallelogram  is  8  yd.  wide  and  3  yd.  high,  what 
is  its  area  ?    What  is  the  area  of  each  of  the  triangles  ? 

6.  Find  the   area  of  a  rectangle  with  base   7  ft.  and 
height  10ft.;  of  a  triangle  with  base  7ft.  and  height  10  ft. 

Notice  that  a  rectangle  is  one  kind  of  a  parallelogram. 

7.  Considering  the    above   examples,  state  a  rule   for 
finding  the  area  of  a  triangle. 

8.  Draw  to  scale  a  triangle  with  sides  6  in.,  7  in.,  and 
8  in.  respectively.    Draw  lines  to  show  that  the  area  of 
the  triangle  is  half  the  area  of  a  rectangle  with  the  same 
base  and  height. 

9.  What  is  the  area  of  a  triangular  garden  with  base 
32ft.  and  height  16ft.? 

Find  the  areas  of  triangles  with  bases  and  heights  as  follows : 

10.  4in.,3.6in.         12.  8yd.,9yd.  14.  9ft.,  4  ft.  4  in. 

11.  9  in.,  7.4  in.        13.  7.5  in.,  8.4  in.        15.  6ft  3  IP...  8ft. 


170  GEOMETRY  OF  SIZE 

Area  of  a  Triangle.    From  the  illustrations  given  and 
the  questions  asked  on  page  169  it  is  easily  seen  that 

The  area  of  a  triangle  is  equal  to  half  the  product  of  the 
base  and  height. 

This  may  be  expressed  by  the  formula 

A  =  i  bh. 

•  4 

For  example,  what  is  the  area  of  a  triangle  of  base  14  in. 
and  height  9  in.  ? 

^  of  14  x  9  sq.  in.  =  63  sq.  in. 

Exercise  26.   Area  of  a  Triangle 

Examples  1  to  9,  oral 

State  the  areas  of  triangles  with  these  bases  and  heights : 

1.  12  in.,  9  in.  4.  28  in.,  3  in.  7.  32  in.,  8  in. 

2.  14  in.,  11  in.          5.  8  in.,  4.5  in.          8.  3.5  in.,  4  in. 

3.  9  in.,  10  in.  6.  3.5  in.,  6  in.          9.  8  in.,  9.5  in. 

10.  How  many  square  yards  of  bunting  are  there  in  a 
triangular  school  pennant  of  base  56  in.  and  height  2  yd.  ? 

Find  the  areas  of  triangles  with  these  bases  and  heights : 

11.  17ft.,  46ft.  14.  36£ft,  17.6ft. 

12.  19.5  ft.,  18.3  ft.  15.  18.3  ft.,  14.4  ft. 

13.  22.7ft,  16.4ft.  16.  29.7yd.,  24.8yd. 

17.  The  span  AB  of  a  roof 
is  40  ft.,  the  rise  MC  is  15  ft., 
the  slope  CB  is  25  ft.,  and  the 
length  BE  is  60  ft.  Find  the 
area  of  each  gable  end  and 
the  area  of  the  roof. 


AREA  OF  A  TRIANGLE 


171 


i> 


18.  On   squared  paper  draw  a  right  triangle  with  the 
two  sides  respectively  1.5  in.  and  2.5  in.    Estimate  the  area 
by  counting  the  squares,  compute  the  area  accurately,  and 
then  find  what  per  cent  the  first  result  is  of  the  second. 

When  we  speak  of  the  two  sides  of  a  right  triangle  we  always 
mean  the  two  perpendicular  sides. 

19.  A  field  65  rd.  by  140  rd.  is  cut  by  a  diagonal  into 
two  equal  right  triangles.   A  railway  runs 

along  this  diagonal  and  takes  3  A.  off 
each  triangular  field.  How  much  is  the 
rest  of  the  field  worth  at  $140  an  acre  ? 

20.  In  this  figure  ABCD  represents  an  8-inch  square, 
E,  F,  G,  and  H  being  the  mid-points  of  the  sides.    In  the 
square  AEOH,  AP  =  QE  =  ER  =  SO 

=  OT  =  •  •  .  =  \AE.  Find  the  area 
of  each  of  the  small  triangles  and 
also  of  the  octagon  PQRSTUVW. 

The  dots  (  •  •  •  )  mean  "  and  so  on  "  and, 
in  this  case,  take  the  place  of  "  UH  =  H  V 
=  WA." 
An  octagon  is  a  figure  of  eight  sides. 

21.  The  triangle  ABC  is  made  by  driving  pins   at  A 
and  B,  running  a  rubber  band  around  them,  and  stretching 
this  band  to  the  point  C.    Now 

imagine  C  to  move  along  CE 
parallel  to  AB,  stopping  first  at 
D  and  then  at  E.  Have  ABC, 
ABD,  and  ABE  different  areas'? 
State  your  reasons  fully. 

Since  any  field  may  be  cut  into  triangles  by  drawing  certain 
diagonals,  the  students  are  now  prepared  to  find  the  area  of  any 
piece  of  land  that  admits  of  easy  measurement. 

JH1 


U         TO 

V 

w 

7           sp~ 
\           \R 

J 

IP         QE 

I 

D 


172  GEOMETRY  OF  SIZE 

Area  of  a  Trapezoid.    If  a  trapezoid  T  has  its  double  cut 
from  paper  and  turned  over  and  fitted  to  it,  like  Z>,  the  two 

together  make  a  parallelo-       , <- -, 

gram.    How  does  the  area     /  T        \          D  / 

of  the  whole  parallelogram   *- — ^ —  ' 


compare  with  the  area  of  the  trapezoid  T?  How  does  the 
base  of  the  parallelogram  compare  with  the  sum  of  the 
upper  and  lower  bases  of  the  trapezoid?  How  do  you  find 
the  area  of  the  parallelogram  ?  Then  how  do  you  find  the 
area  of  the  trapezoid  ?  D  r 

If    from    the    trapezoid    A  BCD,    here 
shown,  the  shaded  portion  is  cut  off  and 
is  fitted  into  the  space  marked  by  the 
dotted  lines,  what  kind  of  figure  is  formed  ?    How  is  the 
area  of  the  resulting  figure  found  ? 

If  the  shaded  portions  of  this  trapezoid 
are  fitted  into  the  spaces  marked  by  the 
dotted  lines,  what  kind  of  figure  is  formed  ?    How  is  the 
area  of  the  resulting  figure  found? 

From  these  illustrations  we  infer  the  following: 

To  find  the  area  of  a  trapezoid,  multiply  the  sum  of  the 
parallel  sides  by  one  half  the  height. 

This  may  be  expressed  by  the  formula 


where  A  stands  for  the  area,  h  for  the  height,  B  for  the 
lower  base,  and  b  for  the  upper  base. 

The  parentheses  show  that  B  and  b  are  to  be  added  before  the 
sum  is  multiplied  by  ^  h. 

'     For  example,  if  h  =  4,  B=7,  and  5  =  5,  we  have 
^=-|-x4x(7  +  5) 
=  2  x  12  =  24. 


AREA  OF  A  TRAPEZOID  173 

Exercise  27.    Area  of  a  Trapezoid 

Examples  1  to  6,  oral 

Find  the  area  of  each  of  the  trapezoids  whose  height  is  first 
given  below,  followed  by  the  parallel  sides  : 

1.  6  in.;  9  in.,  11  in.  7.  18  in.;'  9.5  in.,  27.3  in. 

2.  8  in.;  14  in.,  6  in.  8.  24  in.;  11£  in.,  9}  in. 

3.  12  in.;  4  Jin.,  3^  in.          9.  17  in.;  18  in.,  26  in. 

4.  11  in.;  8  in.,  12  in.  10.  14ft;  6  ft.  4  in.,  9ft. 

5.  9  in.;  4^  in.,  5Jin.  11.  42yd.;  19|yd.,  37|yd. 

6.  13  in.;  11  in.,  7  in.  12.  127ft;  96f  ft,  108J  ft 

13.  Find  the  number  of  acres  in  a  field  in  the  form  of 
a  trapezoid,  the  parallel  sides  being  33^  rd.  and  17^  rd. 
and  the  distance  between  these  parallel  sides  being  14  rd. 

14.  If   the   area   of   a  trapezoid  is   396  sq.  in.  and  the 
parallel  sides  are  19  in.  and  17  in.,  what  is  the  height? 

15.  In   this   figure    show  that   we   may  find   the    area 
of  the  trapezoid   by  adding   the   areas   of  two  triangles. 

This  should  be  taken  up  at  the  blackboard.  b 

The  teacher  should  show  that  in  this  case  we      /\         \ 
have  £  hB  +  %hb  =  -|  h  (B  +  b),  just  as  /         \x  \    \h 


A  little  algebra  may  thus  be  introduced  as  necessity  requires  and 
the  way  made  easier  for  more  elaborate  algebra  later. 

16.  Suppose  that  the  upper  and  lower  bases  of  a  trape- 
zoid are  equal,  does  the  formula  for  the  trapezoid  still 
hold  true  ?  The  trapezoid  becomes  what  kind  of  a  poly- 
gon ?  The  formula  becomes  the  formula  for  what  figure  ? 

Practical  outdoor  work  in  measuring  fields  and  in  computing 
areas  may  now  be  given,  or  it  may  be  postponed  until  after  page  174 
has  been  studied. 


174 


GEOMETRY  OF  SIZE 


K 


Area  of  any  Polygon.  A  polygon  like  ABCDEF  may  be 
divided  into  triangles,  parallelograms,  and  trapezoids  as 
here  shown,  and  the  areas  of  these 
parts  may  be  found  separately  and 
then  added. 

As  an  exercise  the  teacher  may  assign  to 
the  class  the  finding  of  the  area  of  the  field 
here  represented,  the  figure  being  drawn  to 
the  scale  1  in.  =  200  rd. 

Area  found  from  Drawing.  Suppose  that  the  area  of  a 
field  ABC  has  to  be  found,  and  that  there  is  a  large 
swamp  as  indicated  in  the  figure.  In  such  a  case  it  is 
not  easy  to  find  the  height  of  the  triangle ;  that  is,  the  dis- 
tance CD.  The  lengths  of  the  three  c 
sides  may,  however,  be  measured,  and 
then  the  area  may  be  found  by  draw- 
ing the  outline  to  scale  and  measuring 
the  height  of  this  triangle. 

Only  the  drawing  to  scale  is  here 
shown.  If  the  scale  is  1  in.  =  100  rd.,  A  5  ~B 

we  see  that  CD  is  90  rd.,  because  CD  is  0.9  in.  If  AB  is 
100  rd.  the  area  of  the  triangle  is  ^  X  90  x  100  sq.  rd.,  or 
4500  sq.  rd.,  which  is  equal  to  28-J-  A. 

Therefore,  to  find  the  area  of  a  field  from  a  drawing, 

Draw  the  plan  to  scale;  divide  the  plan  into  triangles; 
from  the  base  and  height  of  each  triangle  on  the  plan  com- 
pute the  base  and  height  of  each  triangle  in  the  field;  from 
these  results  find  the  areas  of  the  several  triangles  and  thus 
find  the  area  of  the  field. 

It  must  be  understood  that  surveyors  have  better  methods,  but 
this  method  is  sufficient  for  our  immediate  purposes.  The  immediate 
object  in  view  is  not  to  make  practical  surveyors  tut  to  show  the 
general  power  of  mathematics. 


AREAS 


Exercise  28.  Areas 

1.  This  plan  represents  a  space  150  ft.  long  and  75  ft. 
wide,  with  two    triangular   flower   beds,   in  a   city  park. 
Around  the  inside  of  the  space 

is  a  sidewalk  6^  ft.  wide.  Meas- 
ure the  figure,  determine  the 
scale  used  in  drawing  the  plan, 
and  find  the  area  of  each  of  the 
flower  beds  in  the  park. 

2.  This  map  is  drawn  to  the  scale  1  in.  =  520  mi.    Care- 
fully measure  the  map  and  determine  approximately  the 
length  of  each  side  of  each  state,  and 

then  find  the    approximate   area  of 
each  state. 

The  results  obtained  will  be,  of  course, 
merely  approximate,  since  the  map  is  so 
small.  The  method  is,  however,  the  one 
which  is  employed  in  practical  work  with 
larger  maps. 

3.  Each  side  of  a  brick  building  with  a  slightly  sloping 
roof  is  in  the  form  of  a  trapezoid,  as  here   shown.    The 
building  is  57  ft.  wide,  57  ft.  high  on  the  front, 

and  52  ft.  high  on  the  rear.    On  this  side  there 

are  4  windows  each  4  ft.  wide  and  9  ft.  high 

and  4  windows  of  the  same  width  but  6^  ft. 

high.    If  it  takes  14  bricks  per  square  foot  of 

outside  surface  to  lay  the  wall,  how  many  bricks  will  be 

needed  to  lay  this'  wall,  deducting  for  the  8  windows  ? 

4.  The   sides  of  a  triangular  city  lot  are  respectively 
72  ft.,  60  ft.,  and  48  ft.    Draw  a  plan  of  the  lot  to  the 
scale  of  1  in.  to  12  ft.,  measure  the  altitude  of  the  scale 
drawing,  and  find  the  altitude  and  area  of  the  lot. 


WYOMING 

UTAH 

COLORADO 

176 


GEOMETRY  OF  SIZE 


12'      20'       11'       20' 


25' 


T 

u 

III 

IV 

V 

16 

20' 

32' 

20' 

'  VT 

».!   VIi 

4! 

VIII 

coj     /Vtoi 

OOl    J 

5.  This  sketch  shows  the  plan  of  some  small  suburban 
garden  plots  which  are  offered  for  sale  at  20  $  a  square  foot. 
Find  the  price  of  each  lot. 

6.  In  a  certain  city  Washing- 
ton Street  runs  east  and  west 
and  intersects  Third  Avenue  at 
right  angles.    Using  the  scale 
1  in.  =  100  ft.,  draw  a  plan  of 

the  property  on  the  southeast  corner  from  the  following 
description  :  Beginning  at  the  corner,  run  south  160  ft., 
then  east  75  ft.,  then  north  15  ft.,  then  east  50  ft.,  then 
by  a  slanting  line  to  a  point  on  Washington  Street  100  ft. 
from  the  corner,  and  then  to  the  corner.  Find  the  area  of 
the  plot  and  the  value  at  $2.20  per  square  foot. 

7.  In  order  to  measure  the  distance  AB  across  a  swamp 
some  boys  measure  a  line  CD,  drawn   as  shown  in  the 
figure,   and    find  it  to  be   280  ft. 

& 

long.    They  find  that  DA  =  40  ft. 

and  CB  =  90  ft.,  DA  and  CB  being 

perpendicular   to    CD.    Draw    the 

plan  to  some  convenient  scale  and  determine  the  distance 

AB.    Find  also  the  area  of  the  trapezoid  A  BCD. 

8.  A  swimming  tank  is  60  ft.  long  and  35  ft.  wide.   Draw 
a  plan  to  the  scale  1  in.  =  10  ft.,  determine  the  length  of 
the  diagonal  by  measurement,  and  then  compute  the  num- 
ber of  yards  that  a  student  will  swim  in  swimming  along 
the  diagonal  of  the  tank  eight  times. 

9.  A  lot  has  a  frontage  of  65  ft.  and  a  depth  of  150  ft., 
and  a  path  runs  diagonally  across  it.    Draw  the  plan  to 
scale   and  find,  by  measurement,  the   distance    saved  by 
using  the  path  instead  of  walking  round  the  two  sides 
at  a  distance  of  2  ft.  outside  the  edges  of  the  lot. 


D  280ft. 


AEEAS  177 

10.  Suppose  that  you  have   360  ft.  of  wire  screen  to 
inclose  a  plot  in  which  to  keep  chickens.    If  you  wish  to 
inclose  the  largest  possible  area  in  the  form  of  a  parallelo- 
gram, triangle,  or  trapezoid,  which  form  would  you  use  ? 
Show  by  a  drawing  on  squared  paper  that  the  form  which 
you  choose  incloses  a  larger  area  than  the  others.    Remem- 
ber that  there  are  several  kinds  of  triangles,  several  kinds 
of  parallelograms,  and  several  kinds  of  trapezoids. 

11.  Draw  three  different  triangles,  each  with  base  2  in. 
and  height  1  in.   Find  the  area  of  each.   What  do  you  infer 
as  to  the  equality  of  the  areas  of  triangles  having  equal 
bases  and  equal  heights  ?    Write  the  statement  in  full. 

12.  Upon  the  same   base   of   2  in.  draw  three  different 
parallelograms,  each  having  a  height  1^-  in.    Find  the  area 
of  each  parallelogram.    What  do  you  infer  as  to  the  equal- 
ity of  the  areas  of  parallelograms  on  the  same  base  and 
with  equal  heights?    Write  the  statement  in  full. 

13.  For  computing  the  area  covered  by  1000  ft.  of  a  river, 
some  boys  at  0  wish  to  find  the  width  OB  of  the  river,  as 
here  shown.    They  know  that  the  distance  AB  is  300  ft. 
and  that  the  angle  at  B  is  90°.    Show 

how,  by  sighting  along  YB  and  XA  and 
by  making  certain  measurements,  the 
boys  can  find  the  distance  OB  without 
crossing  the  river. 

14.  Find  the  area  of  an  equilateral 
triangle  3.1  in.  on  a  side. 

This  may  be  done  by  drawing  the  triangle  on  squared  paper  and 
counting  the  squares,  or,  more  accurately,  by  first  approximating 
the  height  by  measurement  on  the  squared  paper. 

15.  Find  the  area  of  an  isosceles  triangle  with  sides 
2  in.,  2  in.,  and  11  in. 


178  GEOMETRY  OF  SIZE 

Exercise  29.   Optional  Outdoor  Work 

1.  Determine  the  area  of  your  school  grounds  by  care- 
fully  making   the   necessary  measurements  and   dividmg 
the  grounds  into  triangles,  if  necessary. 

2.  In   Ex.  1  determine  the  area   by  drawing  the  plan 
to  scale. 

3.  Drive  two  stakes  in  the  ground  at  A  and  j#,  12  ft. 
apart.    Fasten  one  end  of  a  15-foot  line  at  A  and  one  end 
of  a  9-foot  line  at  B.    Draw  the  loose 

ends  taut  and  drive  a  stake  where  they 
meet,  at  C.  What  kind  of  an  angle  is 
formed  at 


4.  Draw  the  figure  of  Ex.  3  to  the  scale 

which  is  four  times  the  one  here  usedv  and  determine  from 
your  figure  some  other  measurements  which  might  be 
used  to  lay  out  the  same  kind  of  angle.  Try  this  on  the 
school  grounds. 

5.  What  is  the  largest  scale  on  which  a  plan  of  your 
school  grounds  could  be  drawn  on  a  piece  of  paper  12  in. 
by  14  in.,  if  you  allow  for  a  margin  of  at  least  1  in.? 

6.  Draw  to  scale  a  plan  of  the  lot  on  which  your  home 
stands  and  indicate  the  ground  plan  of  the  house. 

7.  Draw  to  scale  a  floor  plan  of  some  public  building 
in  your  vicinity.    Compute  the  area  covered. 

8.  Lay  off  on  your  school  grounds  an  isosceles  triangle, 
an  equilateral  triangle,  and  a  right  triangle,  each  with  a 
perimeter  of  30  ft.    Compute  and  compare  the  areas. 

9.  Lay  off  on  your  school  grounds  several  rectangles, 
each   of  which  has  a  perimeter   of   30  ft.    Compute   and 
compare  the  areas. 


ANCIENT  PROOFS 


179 


f/mo  ri'ht  lines  cut  the  one  the  other,  the  hed  angles jklk 
equdl iheone  to  the  othe 


page  from  the  first  English  edition  of  the  great  geometry 
•written  by  Euclid  of  ^Alexandria,  about  300  2?.  £. 
^ow  r/4«  ancient  Gr.eeks prated  their  statements. 


180  GEOMETRY  OF  SIZE 

10.  If  the  street  is  to  be  paved  in  front  of  your  school- 
house,  what  measurements  are  necessary  to  determine  the 
area  to  be  covered  ?    Make  the  measurements  for  the  block 
in  which  your  schoolhouse  stands,  draw  the  plan  to  scale, 
and  compute  the  area. 

11.  If  a  sidewalk  is  to  be  laid  in  front  of  your  school- 
house,  what  measurements  are  necessary  and  what  prices 
must  be  known  in  order  that  you  may  find  the  cost  of  the 
walk  ?    Make  the  measurements,  find  by  inquiry  the  prices, 
and  compute  the  cost  of  the  walk. 

Such  examples  are  merely  typical  of  the  work  which  many  schools 
will  wish  to  have  done.  It  is  impossible,  however,  to  anticipate  the 
practical  cases  which  may  arise  in  any  given  locality.  They  may 
relate  to  some  building  in  process  of  erection,  to  the  laying  of  a 
water  main  in  the  street,  to  the  reservoir  of  the  city  water  supply, 
or  to  the  cost  of  stone  steps  for  a  schoolhouse.  The  important  thing 
is  that  the  problem  should  be  real  and  interesting  to  the  class. 

12.  Compute  the  number  of  square  feet  of  the  surface  of 
some  building  which  needs  to  be  painted,  find  the  average 
cost  per  square  yard  for  painting  it  one  coat,  and  then 
compute  the  cost  of  painting  the  building. 

13.  Suppose  that  a  water  main  is  to  be  laid  in  the  street 
in  front  of  the  schoolhouse.   Ascertain  by  inquiry  the  usual 
width  of  a  trench  for  such  a  purpose,  and  draw  a  plan  of 
the  street  to  scale,  showing  the  location  of  the  trench  and 
giving  it  the  proper  width  to  scale  on  the  drawing. 

14.  In  the  upper  picture  on  the  opposite  page  can  you  see 
how  the  height  of  the  tower  could  be  measured  by  simply 
tipping  the  quadrant  over  flat  and  making  certain  measure- 
ments on  the  ground?    Try  this  plan  in  measuring  the 
height  of  some  tree  or  building. 

In  this  case  also  it  may  be  noticed  that  the  angle  is  exactly  45°, 
and  so  there  is  another  and  better  way  of  finding  the  height. . 


ANCIENT  INSTRUMENTS 


181 


Illustrations  from  old  books  on  geometry,  showing 

hov>  the  height  of  a  tower  or  the  distance  to  an  island  can  be  found 

by  the  aid  of  a  simple  instrument  t>hich  can  easily  be  made. 


182  GEOMETRY  OJ  SIZE 

Ratio.  We  often  hear  of  the  ratio  of  one  number  to 
another,  as  when  some  one  speaks  of  the  ratio  of  the  width 
of  a  tennis  court  to  its  length,  or  the  ratio  of  daylight  to 
darkness  in  the  winter,  or  the  ratio  of  a  man's  expenses 
to  his  income.  By  the  ratio  of  3  to  4  we  mean  3-5-4,  or 
!>  while  the  ratio  of  1  in.  to  1  ft.  is  -j^r  and  the  ratio  of 
\  to  |  is  1  -J-  f ,  or  f . 

The  relation  of  one  number  to  another  of  the  same 
kind,  as  expressed  by  the  division  of  the  first  number  by 
the  second,  is  called  the  ratio  of  the  first  to  the  second. 

A  few  examples  of  ratio  should  be  given  on  the  blackboard.  Tims 
the  ratio  of  $3  to  $6  is  •§>  or,  in  its  simplest  form,  \ ;  the  ratio  of 
1  yd.  to  1  ft.  is  the  same  as  the  ratio  of  3  ft.  to  1  ft.,  or  3  ;  the  ratio 
of  5  to  2  is  -|,  or  2^ ;.  and  the  ratio  of  any  number  to  itself  is  1. 

The  ratio  of  2  to  3  may  be  written  in  the  fraction  forms, 
^  or  2/3,  or  it  may  be  written  with  a  colon  between  the 
numbers ;  that  is,  as  2 :  3. 

The  teacher  should  explain  to  the  class  that  the  ratio  of  12  ft.  to 

12  ft.    12 
4  ft.,  for  example,  may  be  written ' ,  —  >  12  : 4,  or  simply  3.    The 

"X   it.  "I 

word  "ratio  "  is  used  for  each  of  these  forms.  The  expression  12  :  4 
is  read  "the  ratio  of  12  to  4,"  or  "as  12  is  to  4,"  12  and  4  being 
called  the  terms  of  the  ratio. 

Since  any  number  divided  by  a  number  of  the  same 
kind,  as  inches  by  inches  or  dollars  by  dollars,  has  an 
abstract  quotient,  we  see  that 

A  ratio  is  always  abstract,  and  its  terms  may  therefore 
be  written  as  abstract  numbers. 

That  is,  instead  of  labeling  our  numbers,  as  in  2  f t. :  4  ft.,  we  may 
omit  all  labels  and  write  simply  2  :  4,  or  |,  or  |. 

Teachers  should  use  the  familiar  fraction  form  first.  Indeed,  the 
special  symbol  (:)  is  slowly  going  out  of  use  because  it  is  not  neces- 
sary. We  often  see  2  :  3  written  as  2/3  instead  of  ^.  Ratios  are  little 
more  than  fractions  and  may -be  treated  accordingly. 


EATIO 


183 


Exercise  30.    Ratio 

All  work  oral 

1.  Expressed'  in   simplest  form,  what   is  the  ratio  of 
6  to  12?  of  12  to  6? 

When  a  ratio  is  asked  for,  the  result  should  always  be  stated  in 
the  simplest  form  unless  the  contrary  is  expressly  stated. 

2.  What  is  the  ratio  of  $4  to  $12?  of  4ft.  to  12ft.? 

3.  What  is  the  ratio  of  4-J  to  9  ?  of  15  to  7|  ? 

4.  In  the  figure  below,  what  is  the  ratio  of  E  to  Z>? 
What  is  the  ratio  of  E  to  C  ? 

When  we  speak  of  the  ratio  of  E  to  D  we  mean  the  ratio  of  their 
number  values  ;  that  is,  of  1  to  2,  the  ratio  being  ^.  When  we  speak 
of  the  ratio  of  E  to  2  B  we  mean  the  ratio  of  1  to  2  x  4.  This  is  4- 

o 

5.  In  the  figure  below,  what  is  the  ratio  of  E  to 
What  is  the  ratio  of  E  to  A?  of  E  to  &  +  C? 

Referring  to  the  figure,  state  the  following  ratios  : 


6.  E  to  %B. 

7.  D  to  2  A. 

8.  2  E  to  D. 

9.  2  D  to  B. 
10.  <7  to  3  E. 


11.  A  to  E, 

12.  C  to  5. 

13.  D  to  C. 

14.  C  to  ^. 

15.  A  to  2D. 


D      E 


16.  What  is  the  ratio  of  any 
number  to  twice  itself  ? 

17.  What  is  the  ratio  of  a  foot 

to  a  yard  ?    of  an  inch  to  a  foot  ?    of  8  oz.  to  1  Ib.  ?    of 
1  pt.  to  1  qt.?    of  2  qt.  to  1  gal.? 

In  every  such  case  the  measures  must  be  expressed  in  the  same 
units  before  the  ratio  is  found.  Thus  the  ratio  of  1  yd.  to  7  ft.  is  the 
ratio  of  3  ft.  to  7  ft.,  or  of  1  yd.  to  2^  yd.,  either  of  which  is  ^. 


184  GEOMETRY  OF  SIZE 

Proportion.  An  expression  of  equality  between  two  ratios 
is  called  a,  proportion. 

For  example,  $5  :  $8  =  10  ft.  :  16  ft.  is  a  proportion.  This 
proportion  is  read  "  $5  is  to  $8  as  10  ft.  is  to  16  ft."  or  "  the 
ratio  of  |5  to  |8  is  equal  to  the  ratio  of  10  ft.  to  16  ft." 
It  may,  of  course,  be  written  simply  5  :  8  =  10  :  16,  or  J-  —  ^|. 

The  first  and  last  terms  of  a  proportion  are  called  the  extremes; 
the  second  and  third  terms  are  called  the  means.  These  expressions 
are  unnecessary,  however,  in  the  treatment  of  the  subject  in  the 
junior  high  school. 

We  often  have  three  terms  of  a  proportion  given  and 
wish  to  find  the  fourth.  For  example,  we  may  have  the 
proportion  n:  14  =27:  63, 

where  n  represents  some  number  whose  value  we  wish  to 
find.    We  may  write  the  proportion  in  the  more  familiar 

fraction  form,  thus:  07. 

n       —  i 


If,  now,  ^  of  n  is  equal  to  |-|,  we  see  that  n  must 
be  equal  to  14  x  -§-|,  or  6. 

The  teacher  should  show  on  the  blackboard  that  we  need  merely 
multiply  the  two  equal  ratios  by  14,  canceling  as  much  as  possible, 
and  we  have  n  =  6. 

If  we  have  4  :  w  =  12  :  6,  we  may  simply  take  the  ratios  the 
other  way,  and  have  n:  4  =  6  :12,  and  then  solve  as  above. 

The  old  method  of  solving  business  problems  by  ratio  and  pro- 
portion is  no  longer  used  to  any  considerable  extent.  The  subject 
of  ratio  has  a  value  of  its  own,  however,  and  proportion  is  peculiarly 
useful  in  geometry. 

It  is  interesting  to  notice  that  in  any  proportion  of 
abstract  numbers  the  product  of  the  first  and  fourth  terms  is 
equal  to  the  product  of  the  second  and  third  terms. 


PKOPORTION  185 

Exercise  31.   Proportion 

Find  the  value  of  n  in  each  of  the  following  proportions: 

1.  n:18  =  7:9.  3.  7:w=9:72. 

2.  7i:42=13:14.  4.  15:13  =  ra:65. 

5.  A  certain  room  is  24  ft.  by  32  ft.  and  the  width  is 
represented  on  a  drawing  by  a  line  9  in.  long.    How  long 
a  line  should  represent  the  length  ? 

6.  When  a  tree  38  ft.  high  casts  a  shadow  14  ft.  long, 
how  long  is  the  shadow  cast  by  a  tree  64  ft.  high  ? 

In  all  such  cases  the  trees  are  supposed  to  be  in  the  same  locality 
and  perpendicular  to  a  level  piece  of  ground. 

7.  If  a  picture  42  in.  by  96  in.  is  reduced  photographi- 
cally so  that  the  length  is  7-|  in.,  what  is  the  width  ? 

8.  By  means  of  a  pantograph  a  student  enlarges  the 
floor  plans  for  a  house  in  the  ratio  of  8  :  3.    If  the  dining 
room  in  the  original  plans  measures  2-|  in.  by  3  in.,  what 
are  the  dimensions  in  the  enlarged  drawing? 

9.  The   sides  of   a  triangle  are   9  in.,  7  in.,  and  6  in. 
Construct  a  triangle  the  corresponding  sides  of  which  are 
to  the  sides  of  the  given  triangle  as  3:4. 

10.  A  map  is   drawn   to  the   scale   of   1  in.  to   0.8  mi. 
How  many  acres  of  land  are  represented  by  a  portion  of 
the  map  1  in.  square  ? 

1  mi.  =  320  rd.,  and  1  A.  =  160  sq.  rd. 

11.  The  floor  of  a  schoolroom  is  24ft.  by  30ft.    The 
total  window  area  is  to  the  floor  area  as  1:5,  and  the 
6  windows  have   equal   areas,  each  window  being   3-^  ft. 
wide.    Determine  the  height  of  each  window  to  the  nearest 
quarter  of  an  inch. 


186  GEOMETRY  OF  SIZE 

Proportional  Numbers.  Numbers  which  form  a  propor- 
tion are  called  proportional  numbers. 

Similar  Figures.    As  stated  on  page  141,  figures  which 
have  the  same  shape  are 
called   similar  figures  and 
are  said  to  be  similar. 

For  example,  these  two  tri- 
angles are  similar.  Likewise 
triangles  ABC  and  XYZ  on 
page  187  are  similar.  x  A  B 

Proportional  Lines.  The  lengths  of  corresponding  lines 
in  similar  figures  are  proportional  numbers ;  that  is,  corre- 
sponding lines  in  similar  figures  are  proportional. 

For  example,  in  the  above  triangles  XY :  YZ  =  AB  :  BC.  In  two 
circles  the  circumferences  and  radii  are  proportional,  the  circum- 
ference of  the  first  being  to  the  circumference  of  the  second  as  the 
radius  of  the  first  is  to  the  radius  of  the  second. 

Exercise  32.   Similar  Figures 

Examples  1  to  4,  oral 

1.  In  the  above  triangles,  if  XY  is  twice  as  long  as  AB, 
how  does  ZX  compare  in  length  with  CA  ? 

2.  In  the  figure  below  state  two  proportions  that  exist 
among  AB,  AD,  AC,  and  AE. 

3.  In   this  figure,  if  AB  is  -|  of  AD,  C, 
what  is  the  ratio  of  AC  to  AE? 

4.  In  the   same  figure,  if  DE  repre- 
sents the  height  of  a  man  6  ft.  tall,  BC 

the  height  of  a 'boy,  DA  the  length  of  the  shadow  cast  by 
the  man,  and  BA  the  length  of  the  shadow  cast  by  the 
boy,  show  how  to  find  the  height  of  the  boy  by  measuring 
the  lengths  of  the  shadows. 


PROPOKTIONAL  LIKES  187 

5.  If  a  tree  BC  casts  a  shadow  35  ft.  long  at  the  same 
time  that  a  post  YZ  which  is  12  ft.  high  casts  a  shadow 
15  ft.  long,  how  high  is  the  tree  ? 

Suppose  YZ  to  be  the  post,  XY  to  be  its  shadow,  and  A B  to  be 
the  shadow  of  the  tree. 

Since  the  triangles  ABC  and  XYZ  are  similar,  we  may  find  h, 
the  height  of  the  tree,  from  the  proportion 

BC  _  YZ 

AB~XY'  _«e 

or  by  writing  the  values, 

35  ~15' 

7 

9«  • 

whence  h  = 


& 

That  is,  the  tree  is  28  ft.  high. 

6.  If  a  tree  casts  a  shadow  58  ft.  long  at  the  same  time 
that  a  post  8  ft.  high  casts  a  shadow  14  ft.  6  in.  long,  how 
high  is  the  tree  ?    Draw  the  figure  to  scale. 

7.  If  a  telephone  pole  casts  a  shadow  27  ft.  long  at  the 
same  time  that  a  boy  5  ft.  tall  casts  a  shadow  4  ft.  6  in. 
long,  how  high  is  the  pole?    Draw  the  figure  to  scale. 

8.  A  boy  threw  a  ball  directly  upward  and  watched  its 
shadow  on  the  sidewalk.   When  the  ball  began  to  descend, 
the  shadow  of  the  ball  was  at  a  fence  post-  32  ft.  away.   The 
boy  was  4  ft.  6  in.  tall  and  his  shadow  was  2  ft.  3  in.  long. 
How  high  did  the  boy  throw  the  ball  above  the  level  of 
the  ground?    Draw  the  figure  to  scale. 

9.  A  water  tower  casts  a  shadow  87  ft.  8  in.  long  at  the 
same  time  that  a  baseball  bat  placed  vertically  upright  casts 
a  shadow  twice  its  own  length  on  a  level  sidewalk.    Find 
the  height  of  the  water  tower.    Draw  the  figure  to  scale. 


188 


GEOMETRY  OF  SIZE 


10.  This  man  is  holding  a  right  triangle  ABC  in  which 
AB  =  BC.    What  is  the  height  of  the  tree  in  the  picture 
if   the   base   of.  the   triangle   is 

5  ft.  3  in.  from  the  ground  and 
if  AD  is  32  ft.  ? 

This  is  a  common  way  employed 
by  woodsmen  for  measuring  the 
heights  of  trees.  The  man  backs 
away  from  the  tree  until,  holding  the 
triangle  ABC  so  that  AB  is  level,  he 
can  just  see  the  top  of  the  tree  along 
the  side  A  C. 

In  all  problems  involving  heights 
and  distances  the  student  should  estimate  the  result  in  advance. 
This  will  serve  as  a  check  on  the  accuracy  of  the  work. 

11.  In  Ex.  10   suppose  that  a  triangle  is  used  which 
has  AB  equal  to  twice  BC,  that  AD  is  62  ft.,  and  that  the 
point  B  is  5  ft.  7  in.  above  the  ground ;   find  the  height 
of  the  tree. 

12.  A  woodsman  wishes  to  determine  the  distance  from 
the  ground  to  the  lowest  branch  of  a  tree.    He  finds  that 
if  he  places    a  stick  vertically 

in  the  ground  at  a  distance  of 

32  ft.  from  the  tree,  lies  on  his 

back  with  his  feet  against  the 

stick,  and   sights  over  the  top 

of  the  stick,  the   line  of  sight 

will  strike  the  tree  at  the  lowest 

limb,   as   shown   in   the   figure. 

The    woodsman's    eye    is    5  in. 

above  the  ground,  the  distance 

EF,  as  shown  in  the  figure,  is  5  ft  6  in.,  and  the  top  of 

the  stick  is  4  ft.  9  in.  above  the  ground.    Determine  the 

distance  BC  from  the  ground  to  the  lowest  branch. 


PROPORTIONAL  LINES 


189 


13.  A  boy  whose  eye  is  15  ft.  from  the  bottom  of  a 
wall  sights  across  the  top  and  bottom  of  a  stick  8  in.  long 
and  just  sees  the  top 

and  bottom  of  the  wall, 
the  stick  being  held 
parallel  to  the  wall  as 
shown.  If  the  bottom 
of  the  stick  is  18  in. 
from  the  eye,  what  is 
the  height  of  the  wall? 

14.  A  woodsman  steps  off  a  distance  of  30  ft.  from  a  tree, 
faces  the  tree,  and  holds  his  ax  handle  at  arm's  length  in 
front  of  him  parallel  to  the  tree.    His  hand  is  2 7 'in.  from 
his  eye,  and  2  ft.  4  in.  of  the  ax  handle  just  covers  the 
distance  from  the  ground  to  the  lowest  limb  of  the  tree, 
How  high  is  the  lowest  limb  of  the  tree  ? 

This  method  suffices  for  a  fair  approximation  to  the  height. 

15.  Wishing  to  find  the  length  AB  of  a  pond,  some 
boys  choose  a  point  C  in  line  with  A  and  B,  and  at  B  and 
C  draw  lines  perpendicular  to  BC,  and 

draw  AD.  By  measuring  they  then  find 
B  C  to  be  84  ft.,  DE  to  be  112  ft.,  and 
EA  to  be  154  ft.  What  is  the  length 
of  the  pond  ? 

16.  In  Ex.  15  what  other  measurements 
may  be  used  to  find  the  distance  AB? 

17.  If  ^  in.  on  a  map  represents  a  distance  of  375  mi., 
how  many  miles  will  2-£  in.  represent  ? 

18.  If  a  tree  casts  a  shadow  40  ft.  long  when  a  post 
5  ft.  high  casts  a  shadow  6^  ft.  long,  how  high  is  the  tree  ? 

19.  If  1-|  in.  on  a  map  represents  a  distance  of  325  mi., 
how  many  inches  represent  a  distance  of  340  mi.  ? 


190  GEOMETRY  OF  SIZE 

20.  In  one  of  the  upper  illustrations  on  the  opposite 
page  suppose  the  length  of  the  shadow  of  the  post  to  be 
1  ft.  6  in.  shorter  than  the  height  of  the  post,  and  suppose 
the  shadow  of  the  tower  to  be  69  ft.  4  in.  and  the  height 
of  the  post  to  be  5  ft.  2  in.    Find  the  height  of  the  tower. 

21.  In  one  of  the  upper  illustrations  on  the  opposite 
page  there  is  also  shown  a  very  old  method  of  finding  the 
height  by  means  of  a  mirror  placed  level  on  the  ground. 
Can  you  see  two  similar  triangles  in  the  picture?    If  so, 
describe  the  method  by  which  you  could  find  the  height 
of  a  tree  in  this  way. 

22.  Some  members  of  a  class  made  a  right  triangle  with 
one  side  9  in.  and  the  other  side  12  in.   One  of  them  held 
the  triangle  so  that  the  longer  side  was  vertical  and  then 
backed  away  from  a  tree  until  he  could  just  see  the  top 
by  sighting  along  the  hypotenuse.   The  class  then  measured 
and  found  that  the  eye  of  the  observer  was  45  ft.  in  a  hori- 
zontal line  from  the  tree  and  4  ft.  10  in.  from  the  ground. 
Draw  the  figure  to  scale  and  find  the  height  of  the  tree. 

23.  Draw  a  plan  of  the  top  of  your  desk  to  scale,  rep- 
resenting the  length  by  3  in.    What  will  be  the  width  of 
the  drawing,  and  how  can  it  be  found? 

24.  The  extreme  length  of  a  new  leaf  is  2  in.  and  the 
extreme  width  is   1  in.     After  the  leaf  has  grown  1  in. 
longer,  maintaining  the  same  shape,  what  is  its  width  ? 

25.  A  girl  is  making  an  enlarged  drawing  from  a  photo- 
graph of  a  friend.    In  the  photograph  the  distance  between 
the  eyes  is  |^  in.  and  the  length   of  the  nose   is  T^-  in. 
If  the  distance  between  the  eyes  in  the  drawing  is  50% 
more  than  it  is  in  the  original,  what  is  the  length  of  the 
nose  in  the  drawing  ? 


PROPORTIONAL  LINES 


191 


Saptrr  1eJ£e$f  con  Mm  <SeC  Safe,  el  con  if  Sfeccko 


urious  illustrations  from  early  vtorkj  on  geometry  showing 
how  heights  -were  found  by  'very  simple  methods  -which  can  be  used  in 
school  today  ^  and  how  surveyors  proceeded  •with  their  •wor^. 


192  GEOMETRY  OF  SIZE 

Exercise  33.  Optional  Outdoor  Work 

1.  Measure  the  height  of  any  tree,  telegraph  pole,  or 
church  spire  in  the  vicinity  of  the  school  building,  using 
any  convenient  method. 

In  such  cases  it  is  desirable  to  have  the  class  discuss  the  methods 
in  the  class  hour  preceding  the  outdoor  work,  deciding  upon  the 
methods  to  be  used.  It  is  then  a  good  plan  to  separate  the  class  into 
groups,  each  group  using  a  different  method  from  the  others.  The 
results  can  then  be  compared  and,  if  the  methods  and  work  are 
equally  good,  an  average  may  be  taken  as  a  fair  approximation. 

2.  Measure  the  distance  from  one  point  in  the  vicinity 
of  the  school,  preferably  on  the  school  grounds,  to  another 
point  so  situated  that  a  line  cannot  be  run  directly  between 
them.    Use  any  convenient  method. 

In  case  no  such  points  can  be  found,  the  distance  across  the  street 
may  be  measured  without  actually  crossing. 

3.  Making  the  necessary  measurements,  find  the  area  of 
the  school  grounds  or  the  area  of  such  a  portion  as  is 
decided  upon  by  the  teacher  and  the  class. 

In  suburban  or  rural  communities  the  areas  of  fields  may  be 
found.  The  class  should  see  that  it  now  has  mastered  enough 
mathematics  for  finding  the  area  of  any  ordinary  field. 

4.  Ascertain  the  cost  of  a  concrete  sidewalk,  per  square 
foot  or  square  yard,  and  compute  the  cost  of  a  good  side- 
walk in  front  of  the  school. 

In  case  any  excavations  are  being  made  for  buildings  near  the 
school,  and  it  is  feasible  to  have  the  class  make  the  necessary  meas- 
urements, the  amount  of  earth  removed  may  be  computed. 

In  some  schools  this  optional  work  may  be  practicable,  while  in 
others  it  may  not  be.  Teachers  will  have  to  be  guided  by  circum- 
stances in  assigning  the  above  and  similar  exercises. 

The  problems  on  page  193  are  typical  of  those  which  may  be 
considered  for  outdoor  work. 


OUTDOOR  WORK 


193 


5.  To  find  the  distance  across  a  river  measured  from 
B  to  A,  a  point  C  was  so  chosen  by  the  class  that  BC  was 
perpendicular  to  AB  at  B.   Then  a  per- 
pendicular to  B  C  prolonged  was  drawn. 

The   class   sighted  from   C  to  A   and 

placed  a  stake  at  E  where  the  line  of 

sight  from  C  to  A  cut  the  perpendicular 

from  D.  By  measurement  DC  was  then 

found  to  be  168  ft.  and  CB  290  ft.  and  DE  125  ft.    What 

was  the  distance  across  the  river,  measured  from  B  to  A  ? 

6.  In  order  to  determine  the  distance  from  A  to  B  on 
opposite  sides  of  a  hill,  what  measurements  indicated  in 
this  figure  are  necessary  ?  Make  a  problem 

involving  this  principle,  with  reference  to 
two  points  near  the  school,  take  the  neces- 
sary  measurements  out  of  doors,  solve  the 
problem,  and  check  the  work  by  measuring 
the  figure  drawn  to  scale. 

7.  Wishing  to  determine  the  length  AB  of  a  pond,  a  class 
placed  a  stake  at  S,  as  shown  in  the  figure.    The  line  BS 
was  then  run  on  to  B',  121  ft.  from  S,  and 

BS  was  measured  and  found  to  be  253  ft. 

By  sighting  from  B,  the  angle  B  was  marked 

off  on  a  piece  of  cardboard,  and  then  the 

angle  B'  was  made  equal  to  it,  B'A'  being 

thus  drawn  to  a  point  A'  exactly  in  line 

with  A  and  S.   By  measurement  B'A'  was  found  to  be  132  ft. 

Find  the  length  of  AB. 

This  should  be  considered   at   the   blackboard   before  solving. 
Notice  the  advantage  of  using  A'  to  correspond  to  A,  and  B'  to  B. 

8.  Make  a  problem  similar  to  Ex.  7j  take  the  necessary 
measurements,  and  solve. 


194  GEOMETRY  OF  SIZE 

Circumference,  Diameter,  and  Radius.  The  line  bounding 
a  circle  is  called  the  circumference.  Any  line  drawn  through 
the  center  of  a  circle  and  terminated  at  each  end  by  the 
circumference  is  called  a  diameter,  and  any  line  drawn 
from  the  center  to  the  circumference  is  called  a  radius. 

/  Ratio  of  Circumference  to  Diameter,  put  from  cardboard 
( several  circles  with  different  diameters.    Mark  a  point  P  on 
each  circumference,  roll  the  circle  along  a 
straight  line,  and  determine  the  length  of 
the  circumference  by  measuring  the  line  be- 
tween the  points  where  P  touches  it.    In 
each  case  the  circumference  will  be  found  to 
be  approximately  %\  times  the  diameter. 

^The  number  3.1416  is  a  closer  approximation,  but  3f  should  be 
used  unless  otherwise  stated.  ) 

Since  all  circles  have  the  same  shape, 

Any  circumference    _  Any  other  circumference  _  qi 
Diameter  of  its  circle         Diameter  of  its  circle 

A  special  name  is  given  to  this  ratio  3y;  it  is  called 
pi  (written  TT),  a  Greek  letter.  That  is,  c  :  d  =  TT, 

or  c  =  trd. 

Since  the  diameter  is  twice  the  radius,  d  =  2  r ;  therefore 
c  =  2  irr. 

1.  Find    the    circumference    of    a    bicycle    wheel    the 
diameter  of  which  is  28  in. 

We  have     c  =  -rrd  =  ^  x  28  in.  =  22  x  4  in.  =  88  in. 

2.  Find  the  radius  which  was  used  in  constructing  the 
base  of  a  circular  water  tank  24.2  ft.  in  circumference. 

Since  c  =  2  irr,  it  follows  that  r  =  c  -4-  2  TT.    Therefore  we  have 
r  =  24.2  ft.  -4-  (2  x  -^)  =  3.85  ft. 


CIRCLES  •     195 

Exercise  34.   Circles 

Examples  1  to  8,  oral 
State  the  circumferences  of  circles  of  the  following  diameters: 

1.  Tin.  2.  21  in.  3.  42  in.  4.  56  in. 

\ 

State  the  circumferences  of  circles  of  the  following  radii : 

5.  14  in.  6.  Tin.  7.  21  in.  8.  28  in. 

Find  the  circumferences,  given  the  following  diameters : 
9.  68.6  in.     11.  9.38ft.       13.  53.9ft.         15.  128.8ft. 
10.  420  in.      12.  3.01ft.       14.  13  ft.  5  in.    16.  116.2ft. 

Find  the  diameters,  given  the  following  circumferences : 
17.  176  in.      18.  770yd.       19.  3.96ft.         20.  48.4ft. 

Find  the  circumferences,  given  the  following  radii  : 

21.  77  in.        22.  105  in.        23.  1.75  in.         24.  126  in. 

25.  What  is  the  circumference  of  a  56-inch  wheel? 

26.  Given   that  the  inner  circumference   of  a  circular 
running  track  is  half  a  mile,  find  the  diameter. 

27.  A  girl  has  a  bicycle  with  26-inch  wheels.   How  many 
revolutions  will  each  wheel  make  in  going  a  mile  ? 

28.  A  circular  pond   30  ft.  in  diameter  is  surrounded 
by  a  path  6  ft.  wide.    What  is  the  length  of  the  outer 
circumference  of  the  path? 

29.  How  many  revolutions  will  an   automobile   wheel 
37  in.  in  diameter  make  in  going  one  mile  ? 

30.  If  the  cylinder  of  a  steam  roller  is  6.5  ft.  in  diameter 
and  8  ft.  long,  how  many  square  feet  of  ground  will  it 
roll  at  each  complete  revolution  ? 


196  GEOMETRY  OF  SIZE 

Area  of  a  Circle.  A  circle  can  be  separated  into  figures 
which  are  nearly  triangles.  The  height  of  each  triangle  is  the 
radius,  and  the  sum  of  the  bases  is  the  circumference.  If 
these  figures  were  exact  triangles  the  area  of  the  circle  would 


be  -^  X  height  x  sum  of  bases  ;  and,  since  they  are  nearly 
triangles  whose  bases  together  are  the  circumference,  we 
may  say  that  the  area  is  -|  x  radius  X  circumference.    It  is 
proved  later  in  geometry  that  this  is  the  true  area. 
We  may  now  express  this  by  a  formula,  thus: 


Since  c  =  2  TIT  we  may  put  2  trr  in  place  of  <?,  and  have 
A  =  \r  X  2  Trr  =  Trr2. 

The  teacher  should  explain  to  the  class,  if  necessary,  that  the  square 
of  a  number  is  the  product  obtained  by  multiplying  the  number  by 
itself,  and  that  it  is  customary  to  write  32  for  the  square  of  3. 

1.  A  tinsmith  in  making  the  bottom  of  a  tin  pail  draws 
the  circle  with  a  radius  of  5  in.    How  much  tin  does  he 
need,  not  counting  waste  ? 

Since  A  =  Trr2, 

we  have  A  =  ^  x  5  x  5  sq.  in.  =  78$  sq.  in. 

2.  In  order  to  have  an  iron  pillar  capable  of  supporting  a 
certain  weight,  the  cross  section  must  be  50y  sq.  in.    What 
radius  should  be  used  in  drawing  the  pattern  ? 

Since  A  =  Trr2,  we  have  A  •*•  TT  =  r2,  or  oOf  +  z?-  =  rz,  and  so  16  =  r2. 
Because  16  =  4  x  4,  we  see  that  r  =  4. 


AREA  OF  A  CIECLE  197 

Exercise  35.  Area  of  a  Circle 

Examples  1  and  2,  oral 

State  the  areas  of  circles,  given  the  radii  as  follows  : 
1.  7  in.       2.  1  in.       3.  1.4  in.       4.  2.8  in.       5.  56  in. 

6.  If  the  radius  of  one  circle  is  twice  as  long  as  the 
radius  of  another  circle,  how  do  the  areas  of  the  circles 
compare  ?    How  do  the  circumferences  compare  ? 

7.  If  one  circle  has  a  radius  three  times   as   long  as 
the  radius  of  another  circle,  how  do  the  areas  compare  ? 
How  do  the  circumferences  compare  ? 

8.  A  circular  mirror  is  2  ft.  3  in.   in  diameter.     Find 
the  cost  of  resilvering  the  mirror  at  36$  a  square  foot. 

9.  What  is  the  area  of  the  cross  section  of  a  3-inch 
water  pipe?  ^-^ 


3.5  ft. 


10.  Find  the  entire  area  of  a  window  the  lower 
part  of  which  is  a  rectangle  3.5  ft.  wide  and  6  ft.    ^ 
high,  and  the  upper  part  a  semicircle,  or  half  circle,    ^ 
as  shown  in  the  figure. 

11.  In  a  park  there  is  a  circular  lake  of  diameter  120  ft. 
Find  the  number  of  square  yards  of  surface  in  a  walk  5  ft. 
wide  around  the  lake. 

12.  In  the  lake  of  Ex.  11  there  is  a  circular  island  with 
radius  25  ft.    What  is  the  surface  area  of  the  water  ? 

13.  What  is  the  cost,  at  $2.10  a  yard,  of  erecting  a  wall 
around  the  lake  of  Ex.  11  ? 

14.  A  circular  tree  has  a  circumference  of  12ft.  3  in. 
at  a  certain  height  above  the  ground.    What  will  be  the 
area  of  the  top  of  the  stump  made  by  sawing  horizontally 
through  the  tree  at  this  point  ? 


198 


GEOMETRY  OF  SIZE 


Exercise  36.  Volumes 

All  work  oral 
1.  If  C=lcu.  in.,  what  is  the  volume  of  1??    of  A? 


A  B  c 

2.  Find  the  volume  •  of  a  workbox  3  in.  by  4  in.  by  2  in/ 

State  the  volumes  of  solids  of  the  following  dimensions : 

3.  4  in.,  5  in.,  6  in.  5.  2  in.,  3  in.,  10  in. 

4.  3  in.,  3  in.,  7  in.  6.  6  in.,  8  in.,  10  in. 

Rectangular  Solid.  A  solid  having  six  sides,  each  side 
being  a  rectangle,  is  called  a  rectangular  solid.  If  all  the 
sides  are  squares  the  rectangular  solid  is  called  a  cube. 

A  cube  may  be  constructed  from 
cardboard  by  cutting  out  a  diagram 
like  this,  bending  the  cardboard  on 
the  dotted  lines,  and  pasting  the 
flaps.  In  a  similar  way  any  rectan- 
gular solid  may  be  constructed. 


Volume  of  a  Rectangular  Solid.    From  the  exercise  given 
above  we  see  that 

The  volume  of  a  rectangular  solid  is  equal  to  the  product 
of  the  three  dimensions. 

Expressed  as  a  formula,  using  initial  letters,  we  have 
V=lbt. 


VOLUMES  199 

\ 

Exercise  37.  Volumes 

Examples  1  to  9,  oral 
State  the  volumes  of  solids  of  the  following  dimensions : 

1.  3",  4",  6".         4.  7',  2',  3'.          7.  3  yd.,  7  yd.,  2  yd. 

2.  10",  4",  5".       5.  20',  30',  4'.      8.  4  in.,  5  in.,  8  in. 

3.  3",  8",  10".      6.  9",  3",  2".       9.  9  ft.,  4  ft,  3  ft. 

10.  A  cellar  24  ft.  by  32  ft.  by  6  ft.  is  to  be  excavated. 
How  much  will  the  excavation  cost  at  45$  a  load  ? 

Consider  a  load  as  equal  to  1  cu.  yd. 

11.  How  much  will  it  cost,  at  52$  a  cubic  yard,  to  dig 
a  ditch  180  ft.  long,  3  ft.  wide,  and  5  ft.  deep  ? 

12.  The  box  of  an  ordinary  farm  wagon  is  3  ft.  by  10  ft. 
and  the  depth  is  usually  24  in.  or  26  in.    Find  the  contents 
in  cubic  feet  and  cubic  inches  for  each  of  these  depths. 

13.  The  interior  of  a  certain  freight  car  is  36  ft.  long, 
8  ft.  4  in.  wide,  and  7  ft.  6  in.  high.    How  many  cubic  feet 
does  it  contain  ?    If  it  is  filled  with  grain  to  a  height  of 
4|-  ft.,  what  is  the  weight  of  the  grain  at  60  Ib.  to  the 
bushel,  allowing  1£  cu.  ft.  to  the  bushel  ? 

14.  To  what  depth  must  a  tank  5  ft.  wide  and  6  ft.  8  in. 
long  be  filled  with  water  to  contain  120  cu.  ft.  of  water? 

15.  It  is   estimated  that  2-|  cu.  ft.  of   corn   in   the   ear 
will  produce  1  bu.  of  shelled   corn.     How  many  bushels 
of  shelled  corn  can  be  obtained  from  a  crib  10  ft.  by  18  ft. 
by  7  ft.,  filled  with  corn  in  the  ear  ? 

16.  A  cellar  22  ft.  by  30  ft.  by  7  ft.  is  to  be  dug  for  a 
house  on  a  level  lot  62  ft.  by  140  ft.    The  dirt  taken  from 
the  cellar  is  to  be  used  on  the  rest  of  the  lot.    To  what 
depth  will  the  lot  be  filled  if  the  dirt  is  evenly  distributed  ? 


200  GEOMETRY  OF  SIZE 

Cylinder.  A  solid  which  is  bounded  by  two  equal  parallel 

circular  faces  and  a  curved  surface  is  called  a  cylinder. 

Only  the  right  cylinder  is  considered  in  this  book. 

The  two  parallel  circular  faces  are  called 
the  bases,  and  the  distance  between  the  bases 
is  called  the  height  or  altitude  of  the  cylinder. 

Volume  of  a  Cylinder.  Since  we  can  place  1  cu.  in.  on 
each  square  inch  of  the  lower  base  if  the  cylinder  is  1  in. 
high,  we  see  that  if  the  cylinder  is  5  in.  high  we  can  place 
5  times  as  many  cubic  inches.  Hence  we  see  that 

The  volume  of  a  cylinder  is  equal  to  the  area  of  the  bate 
multiplied  by  the  altitude. 

Expressed  as  a  formula,  using  initial  letters,  we  have 

V=bh. 
Since  the  base  b  is  a  circle,  it  is  equal  to  Trr2,  and  so 

V= 


Exercise  38.   Volume  of  a  Cylinder 

1.  What  is  the  capacity  of  a  cylindric  gas  tank  60  ft. 
in  diameter  and  38  ft.  6  in.  high  ? 

In  all  such  cases  the  inside  dimensions  are  given: 

2.  Find  the  capacity  in  gallons  (231  cu.  in.)  of  a  water 
tank  of  diameter  24  ft.  3  in.  and  height  66  ft.  9  in. 

3.  A  water  pipe  16  ft.  long  has  a  diameter  of  1  ft.  9  in. 
How  many  cubic  inches  of  water  will  it  contain  ? 

4.  A  farmer  built  a  silo  24  ft.  high  and  14  ft.  in  diameter. 
If  1  cu.  ft.  of  silage  averages  37  lb.,  how  long  will  the  silage 
that  fills  the  silo  last  52  cows  at  38  lb.  per  cow  per  day  ? 

Such  problems  should  be  omitted  in  places  where  the  students 
are  not  familiar  with  silos. 


CYLINDERS  201 

Curved  Surface  of  a  Cylinder.  If  we  cut  a  piece  of  paper 
so  that  it  will  just  cover  the  curved  surface  of  a  cylinder, 
how  may  we  find  the  area  of  the  paper?  What  are  the 
dimensions  of  a  piece  of  paper  that  will  just  cover  the 
curved  surface  of  a  cylinder  6  in.  high  and  having  a  cir- 
cumference of  8  in.  ?  What  is  the  area  of  the  paper  ?  What 
is  the  area  of  the  curved  surface  of  the  cylinder  ? 

We  see  that  the  area  of  the  curved  surface  of  a  cylinder 
is  equal  to  the  circumference  multiplied  by  the  height. 

That  is,          area  =  circumference  X  height, 
or  area  =  -^2-  x  diameter  x  height. 

Expressed  as  formulas  the  above  statements  become  • 
A  =  ch,     A  =  Trdh,     or     A  =  2  Trrh. 

Exercise  39.    Curved  Surface  of  a  Cylinder 

All  work  oral 

1.  How  many  square  feet  in  the  curved  surface  of  a 
wire  1  in.  in  circumference  and  600  ft.  in  lengtji  ? 

2.  How  many  square  feet  of  tin  in  a  pipe  of  length  12  ft. 
and  circumference  6  in.,  allowing  1  sq.  ft.  for  the  seam  ? 

3.  A  tin  cup  is  4  in.  high  and  7  in.  around,  including 
allowance  for  soldering.    How  many  square  inches  of  tin 
are  needed  for  the  curved  surface? 

4.  A  tin  water  pipe  has  a  circumference  of  9  in.  and  a 
length  of  10  ft.,  both  measures   including  allowance   for 
soldering.    How  many  square  inches  of  tin  were  used  ? 

State  the  areas  of  the  curved  surfaces  of  cylinders  with 
heights  and  circumferences  as  follows : 

5.  36  in.,  80  in.          6.  40  in.,  50  in.          7.  40  in.,  35  in. 


202  GEOMETRY  OF  SIZE 

Exercise  40.   Volume  and  Surface  of  a  Cylinder 

1.  How  many  square  feet  of  sheet  iron  will  be  required 
to  make  a  stovepipe  4|  ft.  long  and  7  in.  in  diameter,  if  we 
allow  Ijr  in.  for  the  lap  in  making  the  seam  ? 

2.  A  certain  room  in  a  factory  is  heated  by  246  ft  of 
steam  pipe  of  diameter  2  in.    Find  the  radiating  surface ; 
that  is,  the  area  of  the  curved  surface  which  radiates  heat. 

3.  Compare  the  surface  of  a  cylinder  12  ft.  long,  having 
a  radius  of  20  in.,  and  the  combined  surfaces  of  10  cylin- 
ders each  12ft.  long  and  each  having  a  radius  of  2  in. 

4.  A  farmer  has  a  solid  concrete  roller  2  ft.  2  in.  in 
diameter  and  6  ft.  wide.    How  many  cubic  feet  of  concrete 
are  there  in  the  roller  ?    How  many  square  yards  of  land 
does  it  roll  in  going  ^  mi.  ? 

5.  At  40$  a  square  yard,  find  the  cost  of  painting  a 
cylindric  standpipe  64  ft.  high  and  9  ft.  3  in.  in  diameter. 

6.  State  a  rule  and  a  formula  for  finding  the  circum- 
ference of  a  cylinder,  given  the  radius,  and  also  a  rule  a,nd 
a  formula  for  finding  the  area  of  the  circular  cross  section. 

7.  State   a  rule   and  a  formula  for  finding  the  total 
surface  of  a  cylinder,  given  the  radius  of  the  base  and 
the  height. 

8.  State  a  rule  and  a  formula  for  finding  the  volume 
of  a  cylinder,  given  the  diameter  and  the  height. 

9.  If  you  know  the  volume  of  a  cylinder  and  the  cir- 
cumference of  the  base,  how  do  you  find  the  height  ? 

10.  A  large  suspension  bridge  has  4  cables,  each  1942  ft. 
long  and  1  ft.  3  in.  in  diameter.  In  letting  the  contract 
for  painting  these  cables,  it  is  necessary  to  know  their 
surface.  Compute  it. 


LATHING  AND  PLASTERING  203 

Lathing  and  Plastering.  Laths  are  usually  4  ft.  long, 
1^  in.  wide,  and  ^  in.  thick  and  are  sold  in  bunches  of 
100.  Since  the  laths  are  laid  ^  in.  apart,  1  lath  is  required; 
for  a  space  2  ft.  long  and  4  in.  wide. 

Metal  lathing  is  also  commonly  used. 

Plastering  is  commonly  measured  by  the  square,  yard, 
and  there  is  no  uniform  practice  in  regard  to,  ^he  method 
of  making  allowance  for  openings.  The  allowance  to  be 
made  should  be  mentioned  in  the  contract. 

Exercise  41.   Lathing  and  Plastering 

1.  How  many  laths  are  required  to  coyer  a  space  16  fts 
long  and  10  ft.  wide  ? 

2.  How  many  laths  are  required  for.  the  walls  and  ceil- 
ings of  the  living  room,  dining  room,  and  kitchen  shown 
in  the  plan  on  page  204  if  the  rooms  are  9  ft.  high  and; 
no  allowance  is  made  for  openings,  baseboards,  or  waste  ? 

3.  At  50  $  per  square  yard,  compute,  the  cost  of  plastering 
the  walls  and  ceilings  of  the  reception  hall,  living  room, 
and  dining  room  shown  on  page  204,  no  allowance  being 
made  for  openings  or  baseboards. 

4.  At  30$  per  square  yard,  compute  the  cost  of  rough 
plastering  the  kitchen  shown  on  page  204,  allowing  for  the 
four  doors,  which   are  7  ft.  high   and  3  ft.  wide   and  for 
the  two  windows,  which  are  6  ft.,  high  and  3  ft.  wide. 

5.  Find  the  total  cost  of  the  materials  required  to  lath 
and  plaster   a   room   at  the  following  prices :    20  bu.   of 
lime  at  45$  per  bushel;  3^  cu.  yd.  of   sand   at   65$  per 
cubic  yard  ;  4  bu.  of  hair  at  45$  per  bushel;  200  Ib.  of 
plaster  of  Paris  at  55$  per  hundred;  2800  laths  at  $2.90, 
per  thousand;  20  Ib.  of  nails  at  16$  per  pound. 


204 


GEOMETRY  OF  SIZE 


Exercise  42.   Practical  Building 

1.  Determine  the  scale  to  which  the  architect  drew  the 
plans  for  a  two-story  frame  house  which  are  shown  below. 


riRST  FLOOR  FLAN 


SECOND  FLOOR  FLAN 


2.  Compute  the  cost,  at  40$  a  cubic  yard,  of  excavating 
for  a  7-foot  cellar  under  the  main  part  of  the  house,  the 
excavation  being  27  ft.  by  34  ft.  6  in. 

3.  Compute  the  cost,  at  62$  a  square  yard,  of  cementing 
the  floor  of  the  cellar  if  it  is  25  ft.  by  32  ft.  6  in. 

4.  Compute  the  cost  of  flooring  with  hard  wood  ^  in. 
thick  the  reception  hall,  living  room,  and  dining  room  at 
$90  per  M  board  feet,  allowing  25%  for  waste. 

A  board  foot  (bd.  ft.)  is  the  measure  of  a  piece  of  lumber  1  ft. 
long,  1  ft.  wide,  and  1  in.  or  less  thick.  The  number  of  board  feet 
in  a  board  less  than  1  in.  thick  is  the  same  as  in  a  board  of  the 
same  length  and  breadth  but  1  in.  thick.  A  fraction  of  a  board 
foot  is  counted  a  whole  board  foot. 

5.  Draw  a  plan  of  the  first  floor  on  a  scale  three  times 
as  large  as  the  one  above,  using  a  pantograph  if  desired. 

6.  Draw  a  plan  of  the  second  floor  on  a  scale  six  times 
as  large  as  the  one  above,  using  a  pantograph  if  desired, 


METRIC  MEASURES  205 

Metric  Measures.  We  are  familiar  with  our  common  meas- 
ures of  length,  including  inches,  feet,  yards,  and  miles ;  of 
weight,  including  ounces,  pounds,  and  tons;  of  capacity, 
including  quarts,  gallons,  and  bushels ;  of  surface,  including 
square  inches,  square  yards,  and  acres.  We  need  to  know 
something,  however,  about  the  measures  used  in  countries 
with  which  we  have  extensive  business  relations.  During 
the  European  war  the  newspapers  spoke  often 
of  the  75-millimeter  guns,  the  305-millimeter 
guns,  the  700-kilogram  shells,  and  the  capture 
of  500  meters  of  trenches.  None  of  this  means 


© 

much  to  us  unless  we  know  what  millimeters, 
kilograms,  and  meters  represent  in  our  measures. 
Our  cities  carry  on  a  great  deal  of  business 
with  foreign  countries,  and  since  those  countries 
buy  our  goods,  we  must  be  able  to  describe  them  in 
terms  of  foreign  measures,  especially  because  of 
the  recent  remarkable  increase  in  our  foreign  trade. 

The  work  on  pages  205-211  is  optional.  It  is  not  re- 
quired in  subsequent  work  in  this  book,  but  it  should 
be  taken  if  time  permits. 

Metric  Measures  of  Length.  A  meter  (m.)  is 
equal  to  39.37  in.,  or  about  1  yd.  3  in.  One  thou- 
sandth of  a  meter  is  called  a  millimeter  (mm.) ;  one 
hundredth  of  a  meter,  a  centimeter  (cm.);  one 
thousand  meters,  a  kilometer  (km.). 

The  meter  and  centimeter  should  be  drawn  on  the 
blackboard. 

The  meter  stick  should  be  used  in  actual  measurement 
in  the  schoolroom.  Students  should  be  told  that  1  km.  is 
about  £  mi. 


Metric  Measures  of  Weight.  A  gram  (g.)  is  equal  to  15.43 
grains,  and  a  kilogram  (kg.)  is  equal  to  1000  g.,  or  2.2  Ib. 


206  GEOMETKY  OF  SIZE 

Metric  Measures  of  Capacity.  A  liter  (1.)  is  nearly  the 
same  as  a  quart.  The  prefix  milli  means  thousandth,  centi 
means  hundredth,  and  kilo  means  thousand,  and  so  we 
know  what  a  milliliter,  centiliter,  and  kiloliter  mean. 

A  liter  is  equal  to  0.91  of  a  dry  quart  or  1.06  liquid  quarts,  but 
such  equivalents  need  not  be  taught  to  students  at  this  time. 

Exercise  43.   Approximations 

All  work  oral 

1.  The  newspaper  says  that  a  75-millimeter  gun  was 
used  by  an  army.   About  what  was  the  diameter  in  inches  ? 

1  m.  -  39.37  in.,  and  75  mm.  =  0.075  of  39.37  in.,  or  about  ^\  of 
40  in.,  or  about  how  many  inches  ? 

2.  A  gun  on  a  battleship  has  a  bore  of  300  mm.   About 
what  is  this  in  inches  ? 

300  mm.  =  0.300  m.  =  0.3  m.  =  about  0.3  of  40  in.,  or  about  how 
many  inches? 

3.  A  manufacturer  shipped  a  lot  of  flatirons  weighing 
3  kg.  each.    Express  this  weight  in  pounds. 

4.  An  order  is  received  in  New  Orleans  for  8000  1.  of 
molasses.    About  how  many  gallons  is  this  ? 

Express  the  following  approximately  in  our  measures : 

5.  2  1.  11.  2  km.  17.  6  m.  23.  £  m.  29.  40  km. 

6.  3  m.  12.  J  km.  18.   2  cm.  24.  24  1.  30.  -40  m. 

7.  1  kg.  13.  4  km.  19.   20  m.  25.   10  1.  31.  40  1. 

8.  51.  14.  8km.  20.   81.  26.   10m.  32.  0.5m. 

9.  5kg.     15.  61.         21.   71.       27.  10kg.    33.  0.5km. 
10.  1  km.    16.  9  1.         22.   7  m.     28.  10  cm.  34.  0.5  1. 


METRIC  MEASURES 


207 


Prefixes  in  the  Metric  System.  Although  we  have  now 
studied  the  most  important  measures  in  the  metric  system, 
we  should  know  thoroughly  the  meaning  of  the  prefixes 
used  and  learn  the  tables  of  length,  weight,  capacity, 
surface,  and  volume.  It  will  be  found  that  the  metric 
system  is  very  simple  if  these  prefixes  are  known. 

Just  as  1  mill  =  0.001  of  a  dollar, 

so  1  millimeter  =  0.001  of  a  meter. 

Just  as  1  cent  =  0.01  of  a  dollar, 

so  1  centimeter  =  0.01  of  a  meter. 

Just  as  the  word  decimal  means  tenths, 
so  1  decimeter  =0.1  of  a  meter. 


PREFIX 

MEANS 

myria- 
kilo- 

10,000 
1000 

hekto- 

100 

deka- 

10 

deci- 

0.1 

centi- 

0.01 

milli- 

0.001 

AS 


WHICH    MEANS 


myriameter     10,000 


kilogram 

hektoliter 

dekameter 

decimeter 

centigram 

millimeter 


1000 
100 
10 
0.1 
0.01 
0.001 


meters 
grams 
liters 
meters 
of  a  meter 
of  a  gram 
of  a  meter 


The  teacher  should  show  that  this  system  is  as  much  easier  than  ^^, 
our  common  one  for  measures  and  weights  as  the  system  of  United 
States  money  is  easier  than  the  English  system.    This  is  the  reason 
why  the  metric  system  is  so  extensively  used  on  the  continent  of 
Europe  and  in  Central  America  and  South  America. 

It  should  always  be  remembered  that  measures  never  mean  much 
to  a  student  unless  they  are  actually  used  in  the  schoolroom. 

The  students  should  be  led  to  see  that  metric  units  may  be 
changed  to  units  of  a  higher  or  a  lower  denomination  by  simply 
moving  the  decimal  point,  exactly  as  in  changing  from  205^  to  $2.05. 

Unless  the  student  is  to  use  the  metric  system  in  his  measure- 
ments in  science,  the  further  study  of  this  subject  may  be  omitted. 


208  GEOMETRY  OF  SIZE 

Table  of  Length.  The  table  of  length  is  as  follows : 

A  myriameter  =  10,000  meters 

A  kilometer  (km.)  =  1000  meters 

A  hektometer(hm.)  =  100  meters 

A  dekameter  10  meters 

Meter  (m.) 

A  decimeter  (dm.)  =  0.1       of  a  meter 

A  centimeter  (cm.)  =  0.01     of  a  meter 

A  millimeter  (mm.)  =  0.001  of  a  meter 

The  meter  is  about  39.37  in.,  3J  ft.,  or  a  little  over  a  yard;  the 
kilometer  is  about  0  6  of  a  mile. 

The  names  of  the  units  chiefly  used  are  printed  in  heavy  black 
type  in  the  tables. 

The  abbreviations  in  this  book  are  recommended  by  various 
scientific  associations.  Some  writers,  however,  use  Km.,  Hm.,  cm. 
and  mm.  for  kilometer,  hektometer,  centimeter,  and  millimeter. 

Exercise  44.  Length 
Express  as  inches,  taking  39.37  in.  as  1  m. : 

1.  35m.          3.  32m.  5.  275cm.         7.  5200mm. 

2.  60m.          4.  47.5m.        6.   4.64cm.        8.  8750mm. 

9.  Express  in  inches  the  diameter  of  a  6-centimeter  gun 
and  the  diameter  of  a  75-millimeter  gun. 

10.  A  certain  hill  is  425  m.  high.    Express  this  height 
in  feet. 

11.  A  tower  is  48.6  m.  high.    Express  this  height  in  feet. 

12.  A  wheel  is  2.1  m.  in  diameter.   Express  this  in  inches. 

13.  The  distance  from  Dieppe  to  Paris  is  209  km.    Ex- 
press this  distance  in  miles. 

14.  The  distance  between   two   places   in   Germany  is 
34.8  km.    Express  this  distance  in  miles. 


METRIC  MEASURES  209 

Table  of  Weight.    The  table  of  weight  is  as  follows : 

A  metric  ton  (t)       =  1,000,000  grams 

A  quintal  (q.)         =  100,000  grams 

A  myriagram           =  10,000  grams 

A  kilogram  (kg.)       =  1000  grams 

A  hektogram  100  grams 

A  dekagram  10  grams 

Gram  (g.) 

A  decigram               =  0.1       of  a  gram 

A  centigram  (eg.)  =  0.01     of  a  gram 

A  milligram  (mg.)  =  0.001  of  a  gram 

A  kilogram,  commonly  cal-led  a  kilo,  is  about  2£  Ib.   A  5-cent  piece 
weighs  5  g.    A  metric  ton  is  about  2204.62  Ib. 

Table  of  Capacity.    The  table  of  capacity  is  as  follows: 

A  hektoliter  (hi.)   =  100  liters 
A  dekaliter  =    10  liters 

Liter  (1.) 

A  deciliter  =      0.1       of  a  liter 

A  centiliter  (cl.)  =      0.01    of  a  liter 
A  milliliter  (ml.)  =     '  0.001  of  a  liter 
A  liter  is  the  volume  of  a  cube  1  dm.  on  an  edge. 

Exercise  45.   Weight  and  Capacity 

Express  as  kilos : 

1.  244  Ib.   4.  120  oz.    7.  2500  g.    10.  28.46  g, 

2.  326  Ib.   5.  68.4  Ib.    8.  6852  g.    11.  5700  g.. 

3.  460  Ib.       6.  400  T.          9.  252.5  Ib.       12.  3268  g, 

Express  as  liters,  taking  1  1.  as  1  qt. : 

13.  5  hi.          15.  3.85  hL       17.  4000ml.       19.  656  hi. 

14.  16  pt.        16.  37|gal.      18.  800  gal.        20.  9.65  hi. 


210  GEOMETRY  OF  SIZE 

Table  of  Square  Measure.  In  measuring  surfaces  in  the 
metric  system  we  use  the  following  table : 

A  square  myriameter  =  100,000,000  square  meters 

A  square  kilometer  (km2.)     =      1,000,000  square  meters 
A  square  hektometer  (hm2.)  =  10,000  square  meters 

A  square  dekameter  100  square  meters 

Square  meter  (m2.),  about  1.2  sq.  yd. 

A  square  decimeter  (dm2.)     =  0.01  of  a  square  meter 

A  square  centimeter  (cm2.)    =  0.0001      of  a  square  meter 
A  square  millimeter  (mm2.)  =  0.000001  of  a  square  meter 

The  abbreviation  sq.  m.  is  often  used  for  m2.,  sq.  cm.  for 
cm2.,  and  so  on. 

In  measuring  land  the  square  dekameter  is  called  an  are 
(pronounced  ar);  and  since  there  are  100  square  dekame- 
ters  in  1  hm2.,  a  square  hektometer  is  called  a  hektare  (ha.). 
The  hektare  is  equal  to  2.47  acres,  or  nearly  2^  acres. 

Exercise  46.    Square  Measure 

1.  Find  the  area  of  a  rectangle  3.2  m.  by  12.7  m. 

2.  Find  the  area  of  a  parallelogram  whose  base  is  45  cm. 
and  height  22.4  cm. 

3.  Find  the  area  of  a  triangle  whose  base  is  7.3  m.  and 
height  4.6  m. 

4.  Find  the  area  of  each  face  of  a  cube  of  edge  27.2  cm. 

5.  A  block  of  granite  is  1.2  m.  long,  0.8  m.  wide,  and 
0.7  m.  thick.    Find  the  area  of  the  entire  surface. 

6.  A  cylinder  has  a  diameter  of  0.75  m.  and  a  height  of 
0.8  m.    Find  the  area  of  each  base  ;  the  area  of  the  curved 
surface ;  the  total  area  of  the  surface. 


METRIC  MEASURES  211 

Table  of  Cubic  Measure.    In  measuring  volumes  in  the 
metric  system  we  use  the  following  table : 

A  cubic  kilometer  =  1,000,000,000  cubic  meters 

A  cubic  hektometer  1,000,000  cubic  meters 

A  cubic  dekameter  1000  cubic  meters 

Cubic  meter  (m3.),  about  1^  cu.  yd. 

A  cubic  decimeter  (dm3.)   =  0.001  of  a  cubic  meter 

A  cubic  centimeter  (cm3.)  =  0.000001        of  a  cubic  meter 
A  cubic  millimeter  (mm8.)=  0.000000001  of  a  cubic  meter 

In  measuring  wood  a  cubic  meter  is  called  a  stere  (st.), 
but  this  unit  is  not  used  in  our  country. 

Exercise  47.  Cubic  Measure 

1.  A  box  is  2.3m.  long,  1.7m.  wide,  and  0.9m.  deep, 
interior  measure.    Find  the  volume. 

2.  What  part  of  a  cubic  meter  is  there  in  a  block  of 
marble  1.3  m.  long,  0.8  m.  wide,  and  0.6  m.  thick  ? 

3.  A  cubic  centimeter  of  water  weighs  1  g.    How  much 
does  a  cubic  decimeter  of  water  weigh  ?    How  much  does 
a  cubic  meter  of  water  weigh  ? 

4.  From  Ex.  3  how  much  does  1  mm3,  of  water  weigh  ? 

5.  Knowing  that  gold  is  19.26  times  as  heavy  as  water, 
find  from  Ex.  4  the  weight  of  1  mm3,  of  gold. 

6.  A  certain  pile  of  wood  is  7  m.  long  and  1.8  m.  high, 
and   the   wood   is   cut  in   sticks   1  m.   long.    How  many 
steres  of  wood  are  there  in  the  pile  ? 

Express  the  following  as  cubic  meters : 

7.  1250  dm3.  9.  2550  cm3.  11.  50,200  dm3. 

8.  257,820mm3.         10.  2700dm3.  12.    75,000cm3. 


212  GEOMETRY  OF  SIZE 

Exercise  48.  Miscellaneous  Problems 

1.  Write  the  formula  for  finding  the  volume  of  a  cuber 
and  also  of  a  rectangular  prism. 

2.  Write   the  formula  for  .finding  the  area  of  each  of 
the  following:  a  circle;  a  trapezoid;  a  triangle;  a  paral- 
lelogram ;  a  rectangle. 

3.  What  must  you  have  given  and  how  do  you  proceed 
to  find  the  area  of  a  trapezoid  ?  the  volume  of  a  cube  ? 
the  area  of  a  circle  ?  the  volume  of  a  cylinder  ?  the  area 
of  the  curved  surface  of  a  cylinder?  the  area  of   a  tri- 
angle ?  the  area  of  the  surface  of  a  cube  ? 

4.  What  must  you  have  given  and  how  do  you  proceed 
to  find  the  number  of  cubic  yards  of  earth  to  be  removed 
in  digging  a  cellar  in  the  shape  of  a  rectangular  solid  ? 

5.  What  must  you  have  given   and  how  do  you  pro- 
ceed to  compute  the  cost  of  polishing  a  cylindric  marble 
monument  at  52$  a  square  yard? 

6.  How  many  loads  of  gravel  averaging  1  cu.  yd.  each 
will  be  required  to  grade  2^  mi.  of  road,  the  gravel  to  be 
laid  15  ft.  wide  and  6  in.  deep  ? 

7.  A  water  tank  is  7  ft.  6  in.  long  and  5  ft.  9  in.  wide. 
Water  is  flowing  through  a  pipe  into  the  tank  at  the  rate 
of  3  cu.  ft.  in  2  min.     How  long  will  it  take  to  fill  the 
tank  to  a  depth  of  3  ft.  8  in.  ? 

8.  At  what  heights  on  the  sides  of  a  cylindric  measuring 
vessel  whose  base  is  8  in.  in  diameter  should  the  marks  for 
1  gal.  (231  cu.  in.),  1  qt.,  2  qt.,  and  3  qt.  be  placed  ? 

9.  Water  is  flowing  into  a  cylindric  reservoir  28  ft.  in 
diameter  at  the  rate  of  280  gal.  a  minute.    Find  the  rate, 
that  is,  the  number  of    inches  per  minute,  at  which  the 
water  rises  in  the  reservoir. 


OUTDOOR  WORK  213 

Exercise  49.    Optional  Outdoor  Work 

1.  Make  the  necessary  measurements;  then  draw  a  plan 
of  your  school  grounds  to  a  convenient  scale.    Indicate  on 
the  plan  the  correct  position  and  the  outline  of  the  ground 
covered  by  the  school  building. 

2.  Draw  a  plan  of  the  athletic  field  where  your  school 
games  are  played.    Indicate  on  the  plan  the  correct  positions 
of  the  principal  features  of  the  field. 

3.  Step  off  distances  of  100  ft.  in  various  directions  on 
your  school  grounds.    Check  the  distance  each  time  before 
stepping  off  the  next.    Do  the  same  for  distances  of  75  ft., 
140  ft.,  and  60  ft. 

4.  Estimate   the   distance   between   two  trees   or   other 
objects  on  your  school  grounds,  then  step  off  the  distance. 
Check  by  measuring  the  distance.   . 

5.  Estimate  the  height  of  several  objects  on  or  near  your 
school  grounds,  and  then  check  the  accuracy  of  your  esti- 
mate by  determining  the  heights  of  the  objects  by  some  of 
the  methods  previously  explained. 

6.  Estimate  the  number  of  square  rods  or  acres  in  your 
schoo1  grounds.    Check  your  estimate  by  measurement. 

7.  Estimate  the  number  of  acres  in  one  of  the  city  or 
village  blocks  near  your  school  or  near  your  home.    Check 
your  estimate  by  measurement. 

8.  Lay  off,  by  estimate,  on  your  school  grounds  an  area 
which  you  believe  is  a  quarter  of  an  acre.     Check  your 
work  by  measurement. 

9.  Estimate  the  extreme  length  and  the  height  of  your 
school  building.    Check  your  estimates.   Which  is  the  more 
difficult  for  you  to  estimate  with  a  fair  degree  of  accuracy, 
length  or  height  ? 


214  GEOMETEY  OF  SIZE 

Exercise  50.   Problems  without  Figures 

1.  If  you  know  the  dimensions  of  a  field  in  rods,  how 
do  you  find  the  area  in  acres  ?   If  you  know  the  dimensions 
in  feet,  how  do  you  find  the  area  in  acres? 

2.  If  you  can  make  all  necessary  measurements  of  a 
triangular  field,  how  can  you  find  the  area? 

3.  If  you  can  make  all  necessary  measurements  of  a 
trapezoidal  field,  how  can  you  find  the  area  ? 

4.  How  can  you  determine  the  width  of  a  stream  with- 
out crossing  the  stream? 

5.  How  can   you   determine   the  height   of  a  church 
spire  without  climbing  to  the  top  ? 

6.  If  you  have  a  good  map  of  the  state  of  Colorado, 
drawn  to  a  scale  which  you  know,  how  can  you  find  from 
the  map  the  number  of  square  miles  in  the  state  ? 

For  our  present  purposes  we  may  consider  that  the  state  is 
rectangular,  although  this  is  only  an  approximation. 

7.  If  you  know  the  circumference  of  a  circular  water 
tank,  how  can  you  find  the  diameter? 

8.  If  you  know  the  circumference  of  a  circle,  how  can 
you  find  the  area? 

Consider  first  the  result  of  Ex.  7. 

9.  If  you   know  the   circumference   and  height   of   a 
cylindric  water  tank,  how  can  you  find  the  capacity  ? 

Consider  Exs.  7  and  8. 

10.  If  you  know  the  length  of  a  water  pipe  in  feet  and 
the  internal  diameter  of  the  pipe  in  inches,  how  can  you 
find  the  number  of  cubic  inches  of  water  it  will  hold  ? 
How  can  you  then  find  the  number  of  gallons  and  the 
weight  of  the  water  it  will  hold  ? 


GEOMETRY  OF  POSITION  215 

III.  GEOMETRY  OF  POSITION 

Position  of  Objects.  We  have  already  seen,  on  page  111, 
that  geometry  can  ask  three  questions  about  an  object: 
(1)  What  is  its  shape  ?  (2)  How  large  is  it  ?  (3)  Where 
is  it  ?  The  first  question  led  us  to  study  those  common 
forms  about  which  everyone  needs  to  know;  the  second 
led  us  to  find  the  size  of  such  forms,  including  not  only 
areas  and  volumes  but  also  heights  and  distances;  and 
now  we  come  to  the  third  question  and  consider  position. 

For  example,  when  two  boys  have  located  the  home 
plate  and  second  base  in  laying  out  a  baseball  field,  how 
can  they  find  the  positions  of  first  base  and  third  base  or 
the  position  of  the  pitcher's  box  ?  Or  if  they  have  located 
the  home  plate  and  the  line  to  the  pitcher's  box,  how  can 
•they  locate  the  other  bases?  Such  questions  show  that 
position  is  an  important  subject  in  geometry. 

It  has  often  happened  in  war  that  people  have  buried 
their  valuables.  They  did  not  dare  to  mark  the  spot, 
because  then  their  enemies  might  find  the  hiding  place 
and  dig  up  the  valuables ;  and  so  it  became  necessary  to 
be  able  to  locate  the  spot  in  some  other  way. 

This  may  be   done   in  various  ways.    For  example,  a 
man  may  select  two  trees,  A  and  B,  160  ft.  apart,  and 
measure  the  distance  from  each  tree  to 
F,  the  point  where  he  buried  his  valu-  y'v 

ables,  say  80  ft.  and  120  ft.,  remember- 
ing that  V  is  north  of  a  line'  drawn     .  • 

A.  B 

from  A  to  B.     Then  when  he  returns 
he  may  take  two  pieces  of  rope  80  ft.  and  120  ft.  long, 
besides  what  is  needed  for  tying,  and  by  stretching  these 
from  the  trees  A  and  B  respectively  he  can  find  Fin  just  the 
same  way  that  the  triangle  on  page  116  was  constructed. 


216  GEOMETRY  OF  POSITION 

Exercise  51.   Fixing  Positions 

1.  During  a  war  a  man  buried  some  valuables.    He  re- 
membered that  they  were  buried  south  of  an  east-and-west 
line  joining  two  trees  60  ft.  apart,  and  that  the  point  was 
50  ft.  from  the  eastern  tree  and  70  ft.  from  the  western  tree. 
Draw  a  plan  to  the  scale  of  10  ft.  =  1  in.  and  indicate  the 
point  where  the  valuables  were  buried. 

2.  In  olden  times,  before  there  were  many  banks,  it  was 
the  common  plan  to  bury  treasures  for  safe-keeping.    Sup- 
pose that  a  man  buried  a  treasure  120  ft.  from  one  tree 
and  160  ft.  from  another  one,  but  that  during  his  absence 
the  second  tree  is  entirely  destroyed.    Draw  a  plan  showing 
where  he  should  dig  a  trench  so  as  to  strike  the  treasure. 

3.  Two  points,  A  and  B,  are  3  in.  apart.    Is  it  possible 
to  find   a  point  which   is   1  in.  from  each  ?    1^  in.  from 
each  ?    2  in.  from  each  ?    3  in.  from  each  ?    Draw  a  figure 
illustrating  each  case  and  consider  whether,  in  any  of  the 
cases,  there  is  more  than  one  point. 

4.  To  keep  the  water  clean  a  farmer  covers  a  spring 
with  a  large  flat  stone  and  pipes  the  water  to  his  house. 
He  then  covers  the  stone  and  piping  with  soil,  and  seeds 
it  all  down.    Before   doing   this   he   takes  the   necessary 
measurements  for  locating  the  spring,  using  as  the  known 
points  two  corners  of  the  field  in  which  it  is.    Draw  three 
plans  showing  how  to  take  the  measurements. 

5.  Two  streets,  each  66  ft.  wide,  intersect  at  right  angles. 
Two  water  mains  meet  40  ft.  from  a  certain  corner  and 
45  ft.  from  the  diagonally  opposite  corner.    Some  workmen 
wish  to  dig  to  find  where  the  mains  meet.    Draw  a  plan 
to   scale  and  show  the   two  possible  places  where  they 
should  dig. 


DRAWING  MAPS  217 

Positions  on  Maps.  Knowing  the  shortest  distance  from 
Chicago  to  Minneapolis  to  be  approximately  354  mi.,  that 
from  Chicago  to  St.  Louis  263  mi.,  that  from  Chicago  to 
Kansas  City  415  mi.,  that  from  Minneapolis  to  St.  Louis 
466  mi.,  and  that  from  Minneapolis  to  Kansas  City  411  mi., 
we  can  easily  make  a  map  showing  the  location  of  all  four 
of  these  cities. 

If  we  draw  the  map  to  the  scale  of  1  in.  =  252  mi.,  the 
map  distance  from  Chicago  to  Minneapolis  will  be  |-|-|in., 
or  1.40  in.,  and  similarly  for  the  t)ther  distances.  That  is, 

Chicago  to  Minneapolis  -||-|-  in.  =  1.4  in. 
Chicago  to  St.  Louis  ||~|  in.  =  1.0  in. 

Chicago  to  Kansas  City  -|^|-  in.  =  1.6  in. 
Minneapolis  to  St.  Louis  2  ill  *n'  =  -^  m* 
Minneapolis  to  Kansas  City  -|^-  in.  =  1.6  in. 

With  the  aid  of  a  ruler  draw  the  line  CM,  making  it' 
1.4  in.  long,  C  representing  Chicago  and  M  representing 
Minneapolis.  Next  place  the  M 

compasses  with  one  point  at 
C  and  draw  an  arc  with  a 
radius  1.0  in.,  representing 
the  distance  to  St.  Louis ; 
also  draw  an  arc  with  radius 
1.6  in.,  representing  the  dis- 
tance to  Kansas  City.  Simi- 
larly, with  center  M  and  radii 
1.8  in.  and  1.6  in.  describe  \ 
arcs  intersecting  the  other  ^\j?  c  ^  ^ 

arcs,  thus  locating  St.  Louis 

and  Kansas  City.  We  can  now  approximate  the  distance 
from  Kansas  City  to  St.  Louis,  for  if  it  is  0.95  in.  on  the 
map,  it  is  really  0.95  of  252  mi.,  or  239  mi. 


218  GEOMETRY  OF  POSITION 

Exercise  52.  Map  Drawing 

1.  The  distance  from  Cincinnati  to  Cleveland  is  222  mi., 
from  Cincinnati  to  Pittsburgh  258  mi.,  and  from  Cleveland 
to  Pittsburgh  115  mi.    Draw  a  line  to  represent  the  dis- 
tance from  Cincinnati  to  Cleveland  and  then  mark  the 
position  of  Pittsburgh.   Use  the  scale  of  1  in.  to  50  mi. 

It  is  to  be  understood  that  the  distances  stated  in  this  and  the 
following  problems  are  only  approximate,  and  that  they  are  meas- 
ured in  a  straight  line  and  not  alocg  roads  or  railways. 

2.  The   distance   from    Philadelphia   to   Harrisburg   is 
95  mi.,  from  Harrisburg  to  Baltimore  70  mi.,  and  from 
Baltimore  to  Philadelphia  90  mi.    Indicate  the  position  of 
the  three  cities,  using  the  scale  of  1  in.  to  75  mi. 

3.  On  a  map  drawn  to  the  scale  of  1  in.  to  108  mi.  the 
distance  from  Atlanta  to  Raleigh  is  3^  in.,  that  to  Savannah 
%  in.,  and  that  to  Jacksonville  2^g-  in.    What  is  the  dis- 
tance in  miles  from  Atlanta  to  each  of  the  other  cities  ? 

4.  On  a  map  of  scale  1  in.  =  145  mi.  the  distance  from 
San  Francisco  to  Portland  is  3|^  in.,  that  to  Seattle  4^  in., 
and  that  to  Los  Angeles  2^  in.    What  is  the  distance  in 
miles  from  San  Francisco  to  each  of  the  other  cities  ? 

5.  On  a  map  drawn  to  the  scale  of  1  in.  to  95  mi.  the 
extreme  length  of  Pennsylvania  is  3  in.  and  the  extreme 
width  is  1|^  in.    What  are  the  dimensions   of  the  state  ? 
What  is  the  approximate  area  of  Pennsylvania? 

6.  The  distance  from  Nashville  to  Memphis  is  2 ^g-  in. 
measured  on  a  map  drawn  to  the  scale  of  1  in.  to  90  mi. 
The  distance  from  Nashville  to  Louisville  is  1^-|-  in.,  from 
Nashville  to  Mobile  4^-  in.,  and  from  Nashville  to  Chatta- 
nooga 1^  in.    What  is  the  distance  in  miles  from  Nashville 
to  each  of  the  other  cities  ? 


LOCATING  POINTS  219 

Points  Equidistant  from  Two  Points.  We  shall  now  con- 
sider again  the  case  of  a  spring  of  water  that  has  been 
covered  by  a  stone  slab  and  earth  so  that  its  position  is 
not  visible.  If  the  farmer  who  owns  the  land  has  lost  the 
measurements  and  remembers  only  that  the  spring  was  just 
as  far  from  one  corner  A  of  the  field  as  from  the  diagonally 
opposite  corner  B,  how  shall  he  dig  to  find  the  spring  ? 

If  A  and  B  are  540  ft.  apart  he  might  try  digging  at 
Jf,  the  mid-point  between  them,  270  ft.  from  both  A  and 
B.  Failing  here,  he  might  try  a  point  P, 
which  is  280  ft.  from  both  A  and  B.  He 
might  then  try  a  point  Q,  which  is  also 
280  ft.  from  both  A  and  B,  and  so  on. 
He  might  thus  try  a  large  number  of 
points,  and  yet  miss  the  spring.  It  would  be  better  for 
him  to  run  a  line  along  the  ground  and  know  that  if  he 
should  dig  along  this  line  he  would  certainly  strike  the 
spring.  Let  us  see  how  this  could  be  done. 

If  you  connect  P,  M,  and  Q,  what  kind  of  line  do  you 
seem  to  have  ?  What  does  this  suggest  as  to  how  the  farmer- 
should  run  the  line  ? 

Take  two  points  A  and  B  on  paper ;  find  four  points  such 
that  each  is  equidistant  from  A  and  B,  and  then  connect 
these  points  and  see  if  you  have  the  kind  of  line  you  expect, 
Was  it  necessary  to  take  as  many  as  four  points  ?  How 
many  points  need  be  taken  ? 

From  the  above  work  we  see  the  truth  of  the  following : 

To  find  all  the  points  ivhich  are  equidistant  from  two  given 
points,  find  any  two  such  points  and  draw  a  straight  line 
through  them.  All  the  required  points  lie  on  this  line  or  on 
the  prolongation  of  this  line. 

Instead  of  using  paper  or  blackboard  for  the  above  work,  points 
may  be  taken  on  the  schoolroom  floor  or  on  the  school  grounds. 

JMl 


220  GEOMETRY  OF  POSITION 

Exercise  53.   Points  Equidistant  from  Two  Points 

1.  Two  water  pipes  are  known  to  join  at  a  place  equi- 
distant from  two  trees  which  are  160  ft.  apart.    One  tree 
is  exactly  north  of  the  other,  and  the  pipes  join  west  of 
the  line  connecting  them.    Draw  a  plan  on  the  scale  of 
1  in.  to  40  ft.  and  show  the  line  along  which  a  workman 
should  dig  to  find  where  the  pipes  join. 

2.  The  residences  of  Mr.  Anderson  and  Mr.  Williams 
are  300  ft.  apart  and  on  the  same  side  of  a  straight  water 
main.   Mr.  Anderson's  residence  is  50  ft.  and  Mr.  Williams's 
residence  is  65  ft.  from  the  water  main.    They  decide  to 
have  trenches  dug  so  as   to  strike  the  main   at  a  point 
equidistant  from  the  two  houses.     Draw  a  plan  on  the 
scale  1  in.  =  60  ft.  showing  the  position  of  the  trenches. 

3.  It  is  planned  to  place  a  circular  flower  bed  in  a  park 
so  that  its  center  shall  be  90  ft.  distant  from  each  of  two 
trees  which  are  80  ft.  apart.    The  radius  of  the  flower  bed 
is  to  be  20  ft.    Draw  a  plan  on  the  scale  of  1  in.  to  20  ft. 

4.  How  can  you  find  a  point  on  one  side  of  your  school- 
room -which   is    equidistant  from   two   of  the  diagonally 
opposite  corners  ?    Draw  a  plan  of  your  schoolroom  on 
the  scale  of  1  in.  to  10  ft.  and  locate  the  point  on  the  side 
of  the  room. 

5.  A  landscape  gardener  wishes  to  plant  a  tree  which 
shall   be    equally  distant   from   three    trees   forming   the 
vertices  of  a  triangle,  as  shown  in  the  figure.  c 
How  can  he  find  where  to  plant  the  fourth 

tree?   Suppose  it  is  known  that  AB  =  140  ft., 

#(7  =  110  ft,  and  CA  =130  ft,  draw  a  plan    A 

on  the  scale  of  1  in.  to  20  ft.  and  determine  the  location  of 

the  fourth  tree  and  its  distance  from  each  of  the  other  trees. 


LOCATING  POINTS  221 

6.  Does  a  point  equally  distant  from  the  three  vertices 
of  a  triangle  always  lie  inside  the  triangle  ?   Consider  these 
triangles :  AB  =  4  in.,  BC=7  in.,   CA  =  Q  in. ;  AB  —  3  in., 
BC=4:  in.,  CA  =  5  in. ;  AB  =  2  in.,  BC  =  3  in.,  CA  =  1.5  in. 
Construct  the  triangles  and  locate  the  points. 

7.  In   a  park   there   are   three   walks   which  meet   as 
shown  in  the  figure,  and  ^45  =  180  ft,  J?(7=125  ft,  and 
CA  =  145  ft,    inside    measure.     It    is 

planned  to  place   a  cluster   of    lights 

which    shall   be    equally  distant   from 

the  three  corners  A,  B,  and  C.    Draw  a 

plan  to  some  convenient  scale,  indicate 

the  position  at  which  the  cluster  of  lights  is  to  be  placed, 

and  determine  its  distance  from  each  of  the  three  corners. 

8.  In   an   athletic  park   are   three   trees  A,  B,  and  Cy 
so  situated  that  AB  =  150  ft,  BC  =  115  ft,  and  CA  =  90  ft 
A   running  track  is  planned,  as  shown  in 

the  figure,  so  that  an  arc  of  a  circle  shall 
pass  close  to  the  trees.  Draw  a  plan-  to 
any  convenient  scale  and  determine  the 
radius  of  the  circle. 

9.  In   the  ruins   of  Pompeii  a  portion   of   the  tire   of 
an  old  chariot  wheel  was  found.    The  curator  of  a 
museum    wished    a    drawing    of   the    original    wheel. 
How  was  it  possible  to  determine  the  center  and  to 
reproduce  the  tire  of  the  wheel  in  a  drawing? 

10.  Three   points,   A,  B,   and.  (7,   are    so    situated    that 
AB  =  3.5  in.,  BC=  2.5  in.,  and  CA  =  4.5  in.    By  drawing 
to  scale  and  measuring,  determine  the  length  of  the  radius 
of  the  circle  which  passes  through  the  three  points. 

11.  Solve  Ex.  10  for  three  points  which  are  so  situated 
that  AB=4  ft  2  in.,  BC=  4  ft  8  in.,  and  CA=  5  ft  1  in. 


222 


GEOMETKY  OF  POSITION 


Road 


O 


Distance  of  a  Point  from  a  Line.  In  the  figure  below, 
if  H  represents  a  house  and  AB  a  straight  road,  how  far 
is  it  to  the  house  from  the  road  ? 

It  is  obvious  that  the  answer  to  the  question  depends 
on  how  we  are  to  go.  If  there  are  several  straight  paths 
such  as  HA,  HB,  HC,  and  HD  leading 
to  the  house  from  the  road,  the  length 
of  each  path  is  an  answer  to  the  ques- 
tion. We  see  at  once,  however,  that  A/ 
the  shortest  of  the  paths  shown  in  the 
figure  is  JIB,  and  when  we  speak  of  the  distance  from  the 
house  to  the  road  or  from  a  point  to  a  line,  we  mean  the 
shortest  distance. 

If  a  stone  P  is  hanging  by  a  string  and  the  other  end  of 
the  string  is  held  at  0,  and  if  the  stone  swings  so  as  just 
to  graze  the  line  AB,  it  is  evident  that  OP 
represents  the  shortest  distance  from  0  to 
the  line  and  also  that  OP  makes  two  right 
angles  with  AB.  These  two  right  angles  are 
BPO  and  OP  A,  and,  of  course,  are  equal.  A 

Perpendicular.  There  is  a  special  name  for  the  straight 
line  which  represents  the  shortest  distance  from  a  point 
to  a  line.  In  the  above  figure  OP  is  said  to  be  perpen- 
dicular to  AB  or  to  be  the  perpendicular  from  0  to  AB. 

The  perpendicular  from  a  point  to  a  line  makes  right  angles 
with  the  line  and  is  the  shortest  patli  from  the  point  to  the  line. 

Perpendicular  must  not  be  confused  with  vertical.    A  plumb  line 
(a   line   with    a   weight    at    the 
end)  hangs  vertically,  but  a  line 
maj  be  perpendicular  to  another 

line  and  still  not  be  vertical.    In i — ~     -r- k"""" B 

each  of  these  figures  OP  is  per- 
pendicular to  AB,  but  in  only  the  first  of  the  figures  is  OP  vertical. 


DISTANCE  FROM  A  LINE  223 

Exercise  54.    Distance  of  a  Point  from  a  Line 

1.  Draw  a  perpendicular  to  the  line  AB  from  the  point 
0  outside  the  line,  proceeding  as  follows:  Set  one  point 
of  the  compasses  at  0,  then  adjust  the  compasses  until  the 
other  point  just  grazes  the  line  AB  at  P.    Draw  OP ;  then 
OP  is  evidently  perpendicular  to  AB. 

See  the  second  figure  on  page  222. 

This  method  of  drawing  the  perpendicular  shows  clearly  what 
the  perpendicular  is,  but  it  is  not  frequently  used  in  practice.  Usually 
the  best  method  is  that  which  is  described  on  page  121 
or  that  on  page  122. 

2.  In  this  figure  AB  is  45  ft.,  and  AC  and 
BC  are  each  52  ft.    Find  by  measurement  the 
perpendicular  distance  CM.  •&      M 

3.  A  straight  gas  main  runs  from  one  street  light  A 
to  another  light  B,  the  distance  AB  being  220  ft.     It  is 
desired  to  place  a  third  lamp   C  in  a  park  fronting  on 
the  street,  so  that  the  distance  AC  shall  be  260  ft.  and  the 
distance  BC  shall  be  190ft.    The  gas  main  AB  is  to  be 
tapped  at  the  point  nearest  to  C. 

Draw  the  figure  to  the  scale  of  1  in.  to  40  ft. ;  find  by 
measurement  the  approximate  distance  of  C  from  AB  and 
the  distance  from  B  to  the  point  where  the  main  is  tapped. 

The  figure  is  somewhat  like  that  of  Ex.  4. 

4.  In  this  figure  ABC  is  an  equilateral  triangle  each  of 
whose  sides  is  14  in.    Draw  the  triangle  to  a  convenient 
scale    and    determine    the    length   of  the  c 
perpendiculars  drawn  from  A  to  BC  and 

from  B  to  AC.  Does  the  perpendicular 
drawn  from  C  to  AB  appear  to  pass 
through  the  point  in  which  the  other  two 
perpendiculars  intersect  ? 


224  GEOMETRY  OF  POSITION 

5.  A  stone  is  tied  to  a  string  15  in.  long.    The  string  is 
held  at  the  other  end,  and  the  stone  is  then  swung  so  that 
it   just   grazes  the  ground.    When 

the  stone  is  at  the  point  2?,  12  in. 
from  the  perpendicular  OA,  how 
far  is  it  from  the  ground  ?  Draw 
the  figure  carefully  to  some  con- 
venient scale  and  thus  find  this  distance  by  measurement. 

6.  Some  boys  stretch  cords  from  the  top  of  a  flagstaff 
to  two  points  on  level  ground  40  ft.  apart. 

The  first  cord  is  90  ft.  long  and  the  second 
is  65  ft.  long.  Draw  the  figure  to  scale  and 
thus  find  the  height  of  the  flagstaff. 


7.  Two  boys  observe  a  bird  on  the  top  of  a  90-foot  flag- 
staff.   One  boy  is  65  ft.  from  the  foot  of  the  staff  and  the 
other  is  130  ft.  from  its  foot.    Draw  the  figure  to  scale  and 
thus  find  the  distance  of  each  boy  from  the  bird. 

8.  Two  trees  on  the  same  bank  of  a  straight  river  are 
285  ft.  apart.    The  distance  from  one  of  the  trees  to  a  point 
on  the  opposite  bank  is  200  ft.  and  the  distance  from  the 
other  tree  to  the  same  point  is  260  ft.    Draw  the  figure 
to  scale  and  thus  find  the  width  of  the  river. 

9.  An  airplane  is  2800  ft.  above  a  level  railway  line 
connecting  two  villages  A  and  J3,  which  are  1.5  mi.  apart. 
The  airplane  is  above  a  point  which  is  ^  of  the  distance 
from  A  to  B.    Draw  the  figure  to  scale  and  thus  find  the 
distance  of  the  airplane  from  each  of  the  villages. 

10.  A  pyramid  has  a  square  base  120  ft.  on  a  side.  The 
distance  from  the  vertex  to  the  mid-point  of  one  side  is 
90  ft.  Draw  the  figure  to  scale  and  thus  find  the  height 
of  the  pyramid. 


DISTANCE  FROM  A  LINE  225 

Points  at  a  Stated  Distance  from  a  Line.  If  a  man  wishes 
to  build  a  house  at  a  distance  of  100  ft.  from  a  straight 
road,  he  can  build  it  in  any  one  of  a  great  many  places, 
for  he  can  stand  anywhere  011  the  road,  lay  off  a  per- 
pendicular on  either  side  of  it,  and  then  measure  a  dis- 
tance of  100  ft.  along  this  perpendicular,  any  point  thus 
found  being  100  ft.  from  the  road. 

Pages  121  and  122  may  profitably  be  reviewed  at  this  time. 

To  illustrate  this  further  we  may  draw  a  straight  line 
XY,  using  a  carpenter' s  square  or  a  right  triangle  to  lay 
off  four  points  on  one  side  of  the  line 
and  ^  in.  from  it  and  to   lay  off  four 

points  on  the  other  side  of  the  line  and     x Y 

^in.    from   it.     It   is    evident  that  the 

points  lie  in  two  lines  each  of  which  is 

^  in.  from  XY.    All  points  on  the  paper  ^  in.  above  or 

below  XY  evidently  lie  on  one  of  these  two  lines. 

In  geometric  and  mechanical  drawing  it  is  necessary  to 
be  able  to  locate  such  points  quickly  and  accurately. 

To  draw  a  line  perpendicular  to  a  given  line  XY,  we 
may,  as  stated  on  page  121,  place  a  right  triangle  ABC  so 
that  one  side  BC  lies  along  XY  and  the  hypotenuse  AC 
along  av  ruler  MN.  Then  AB  is  perpendicular  to  XY,  and 
as  we  slide  the  triangle  along  MN,  as  shown  in  the  figure, 
AB  will  remain  perpendicular  to  XY  and  the  line  can  be 
drawn  with  accuracy  right  across  XY. 

After  we  have  drawn  two  such  perpendiculars  on  each 
side  of  XY,  and  the  required  distance  has  been  laid  off,  we 
can  draw  the  two  lines  which  contain  all  the  points  at  the 
given  distance  from  the  line.  It  is  evident  that  we  need 
to  find  only  two  points  of  a  straight  line  in  order  to  draw 
the  line  with  a  ruler. 


226  GEOMETEY  OF  POSITION 

Exercise  55.   Points  at  a  Given  Distance  from  a  Line 

1.  Draw   a  plan   showing  40  ft.  of  a  straight  railroad 
track  4  ft.  8^-  in.  wide,  using  the  scale  of  1  in.  to  8  ft.  and 
being  careful  to  make  the  two  rails  everywhere  equally 
distant  from  each  other. 

2.  Using  a  convenient  scale,  draw  a  plan  of  the  straight- 
away of  a  running  track  showing  100  yd.  of  a  straight 
track  16  ft.  wide. 

3.  Draw  a  plan  of  the  floor  of  your  schoolroom,  being 
careful  to  make  the  sides  perpendicular  at  each  corner. 
If  the   opposite  sides  are  parallel   you  can  test  whether 
your  schoolroom  is  a  perfect  rectangle  by  measuring  the 
diagonals  to  find  whether  they  are  equal. 

4.  An  old  description  of  a  corner  lot  states  that  there 
is  a  covered  well  30  ft.  from  one  of  two  streets  which  cross 
at  right  angles  (but  it  does  not  state  which  street)  and 
50  ft.  from  the  other  street.    Draw  a  plan  to  a  convenient 
scale  and  indicate  all  possible  positions  of  the  well. 

5.  The  residence  of  Mr.  Weber  is  130  ft.  from  a  straight 
road.    He  wishes  to  build  a  garage  80  ft.  from  his  house 
and  120  ft.  from  the  road.    Indicate  on  a  plan  the  position 
of  the  house  and  the  garage. 

6.  Two  straight  driveways  in  a  park  meet  as  shown  in 
the  figure.   A  fountain  is  to  be  placed  80  ft.  from  driveway 
AB  and  60  ft.  from  driveway  CD.    Draw  a 

figure  to  scale  showing  the  two  possible 
positions   of  the  fountain   and  determine 

.  A.  B 

the  distance  between  these  positions.   First 
copy  the  figure  accurately,  enlarging  it  on  any  convenient 
scale,  extending  it  as  may  be  necessary,  and  taking  the 
width  of  driveway  AB  to  be  40  ft. 


POSITION  FIXED  BY  TWO  LINES  227 

Position  fixed  by  Two  Lines.  Having  already  seen  how 
the  position  of  an  object  can  be  found  if  we  know  its  dis- 
tance from  each  of  two  points,  let  us  see  if  we  can  find  the 
position  if  we  know  its  distance  from  each  of  two  lines. 

Take,  for  example,  your  desk.  If  anyone  was  told  that 
your  desk  is  6  ft.  to  the  west  of  the  east  wall  and  8  ft. 
north  of  the  south  wall,  could  he  find  which 
desk  is  yours?  How  would  he  do  it?  In 
this  plan  he  might  run  a  line  6  ft.  from  BC 
and  parallel  to  it,  and  another  line  8  ft.  from 
AB  and  parallel  to  it,  and  where  these  lines 
crossed  he  would  find  the  desk  Z>.  A 

This  is  the  way  we  locate  a  place  on  a  map  of  the 
world.  We  say  it  is  so  many  degrees  north  or  south  of 
the  equator  and  so  many  degrees  east  or  west  of  the 
meridian  of  Greenwich.  Thus  we  say  that  a  place  is 
40°  N.  and  70°  W.,  meaning  that  it  is  40°  north  of  the 
equator  and  70°  west  of  the  prime  meridian,  the  one 
which  passes  through  Greenwich. 

The  principle  applies  whether  we  use  a  Mercator's  projection  or  a 
globe,  except  that  the  curvature  of  the  lines  is  seen  in  the  latter  case. 

The  lines  we  use  need  not  meet  at  right  angles.  For 
example,  if  we  know  that  two  water  mains  join  at  a  point 
60  ft.  south  of  the  road  AB,  as 
here  shown,  and  90  ft.  from  the 
road  CD  and  on  the  side  towards 
J5,  we  can  locate  the  point  P  by  A'. 
simply  drawing  two  lines.  How 
are  these  lines  drawn  ? 

Tf  we  know  that  P  is  60  ft.  south  of  AB  and  90  ft.  from  CD, 
but  do  not  know  on  which  side,  there  would  be  two  possible  points. 
Under  what  circumstances  would  there  be  three  possible  points, 
or  four  possible  points? 


228  GEOMETRY  OF  POSITION 

Exercise  56.   Position  fixed  by  Two  Lines 

1.  Suppose  that  we  do  not  know  the  side  of  the  road  CD 
on  which  the  two  water  mains  mentioned  on  page  227  join, 
but  we  do  know  that  it  is  on  the 

south  side  of  the  road  AB.  Draw 
a  figure  showing  all  the  possible 
positions  of  the  water  mains. 

2.  Suppose  that  we  know  that 

the  water  mains  join  to  the  west  of  (7Z>,  but  do  not  know 
whether  the  point  is  to  the  north  or  to  the  south  of  AB. 
Draw  a  figure  showing  all  the  possible  positions. 

3.  Suppose  that  we  are  uncertain  as  to  the  side  of  each 
road  on  which  the  water  mains  join.    Draw  a  figure  show- 
ing all  the  possible  positions. 

4.  Each  side  of  a  square  park  ABCD  is  150  yd.  long. 
A  monument  is  to  be  erected  in  the  park  at  a  point  130  ft. 
from  AB  and  170  ft.  from  BC.    Draw  the  figure  to  scale 
and  determine  the  distance  of  the  monument  from  each 
corner  of  the  park. 

5.  There  are  three  survivors  of  a  shipwreck.    The  first 
says  that  the  ship  lies  between  2  mi.   and  2-|  mi.  from  a 
straight  coast  line  which  runs  from  a  lighthouse  L  to  the 
west;  the  second  says  that  the  ship  lies  between  l^mL 
and  2  mi.  from  a  straight  coast  line  which  runs  from  L  to 
the  north ;   and  the  third  says  that  it  lies  2^  mi.  from  L. 
Can  they  all  be   right?    If  so,  draw  the  figure  to  scale 
and  indicate  where  to  dredge  for  the  wreck. 

6.  In  a  rectangular  field  ABCD,  90  rd.  by  130  rd.,  there 
is  a  water  trough  which  is  290  ft  from  the  side  AB  and 
320  ft.  from  BC.    Draw  the  figure  to  scale  and  thus  find 
the  distance  of  the  water  trough  from  the  point  D. 


POSITION  FIXED  BY  TWO  LINES  229 

Points  Equidistant  from  Two  Lines.  Suppose  that  two 
roads  AB  and  CD  intersect  at  0  and  that  it  is  desired 
to  place  a  street  lamp  at  a  point 
equidistant  from  the  two  streets. 
How  many  possible  positions  are 
there  for  the  lamp  ? 

If  we  think  of  ourselves  as  walk- 
ing  in  such  a  direction  as  to  be 
always  equidistant  from  OB  and  OD,  we  see  that  we  shall 
be  walking  along  ON,  Similarly,  to  be  equidistant  from 
OD  and  OA  we  must  walk  along  OP.  In  general,  any 
point  on  any  of  the  dotted  lines  in  the  figure  is  equi- 
distant from  AB  and  CD.  Therefore  we  may  locate  the 
lamp  anywhere  on  either  line,  these  lines  evidently  bisect- 
ing the  angles  formed  by  the  roads. 

Exercise  57.    Points  Equidistant  from  Two  Lines 

1.  By  using  this  figure  draw  a  line  containing  points 
equidistant  from  two  lines.  ,        D 

The  dotted  lines  are  each  ^in.  from  AB          * ^ ^ — 

and  CD  respectively.     They  intersect  at  P, 
and  OP  is  drawn  and  prolonged  to  M. 

Z.  Draw    a    line    containing    points 
equidistant  from  two  lines  by  using  a  figure  somewhat  like 
the  one  used  in  Ex.  1,  page  228. 

3.  In  a  park  P,  which  lies  between  two  streets  M  and 
JV,  as  shown  in  the  figure,  an  electric 
light  is  to  be  placed  so  as  to  be  equidis- 
tant from  M  and  N  and  80  ft.  from  the 
corner  C.  Copy  the  figure  and  show  how 
to  find  the  position  of  the  light. 

In  this  case  a  circle  intersects  a  straight  line. 


230  GEOMETRY  OF  POSITION 

4.  The  manager  of  an  amusement  park  decides  to  place 
six  lights  at  intervals  of  80  ft.,  so  that  each  light  .shall 
be  equidistant  from  two  intersecting  drive- 
ways M  and  N,  as  shown  in  the  figure.    The 

first  light  is  to  be  placed  15  ft.  from  the  inter- 
section of  the  driveways.  Copy  the  figure  to 
scale  and  indicate  the  position  of  each  of  the  six  lights. 

5.  A  fountain  is  to  be  erected  in  the  space  S  between 
the  two  streets  M  and  N.    The  contract 

provides  that  the  fountain  shall  be  50  ft. 
from  the  nearest  side  of  each  of  these 
streets.  Copy  the  figure  and  indicate 
the  position  of  the  fountain. 

6.  A  monument  is  to  be  so  placed  in  a  triangular  city 
park  that  it  shall  be  equally  distant  from  the  three  sides. 
Draw  a  plan  on  any  convenient  scale  and 

show  on  the  plan  all  the  lines  necessary  to 
find  the  point  0.  Is  0  equidistant  from  the 
sides  AB  and  BCt  Is  it  equidistant  from 
AB  and  CA?  Is  it  necessary  to  draw  a  line  from  (7? 

7.  A  flagpole  is  to  be  so  placed  that  it  shall  be  equally 
distant  from  the  three  sides  of  a  triangular  park  whose 
sides  are  380  ft.,  270  ft.,  and  300  ft.  respectively.    Draw 
a  plan  on  a  convenient  scale  and  find  the  distance  of  the 
flagpole  from  each  side  of  the  park. 

8.  A  water  main   has  a  gate  located  at  a  point  7  ft. 
from  a  certain  lamp-post  which  stands  on  the  edge  of  a 
straight  sidewalk.    The  gate  is  placed  3  ft.  from  the  edge 
of  the  walk,  towards  the  street.    Draw  a  plan  showing 
every  possible  position  of  the  gate. 

9.  Consider  Ex.  8  when  the  gate  is  located  at  a  point 
3  ft.  from  the  lamp-post. 


USES  OF  ANGLES  231 

Use  of  Angles.  There  is  still  another  method  by  which 
a  man  may  locate  valuables  which  he  has  buried.  Suppose 
as  before  that  there  are  two  trees,  A  and 
B,  and  that  he  has  buried  his  valuables 

at  X.     If  he   knows  the   exact  direction    A*— B 

and  distance  from  A  to  X,  he  can  easily  find  the  place 
where  the  valuables  are  buried. 

Both  the  distance  and  direction  can  be  recorded  on  paper. 
For  example,  the  man  might  write :  "  30°  north  of  line  joining 
the  trees,  150  ft.  from  the  west  tree,"  and  this  would  recall 
to  his  mind  that  the  angle  of  30°  is  to  be  measured  at  A, 
namely,  the  angle  BAX,  and  that  X  will  be  found  150  ft. 
from  A.  The  man  can  lay  off  the  angle  on  a  piece  of 
paper  by  the  aid  of  a  protractor  like  the  one  described  on 
page  115,  and  can  then  sight  along  the  arms  of  this  angle. 

The  use  of  the  protractor,  as  given  on  page  115,  may  be  reviewed 
at  this  time  if  necessary. 

Exercise  58.   Uses  of  Angles 

1.  Estimate  the  number  of  degrees  in  each  of  the  following 
angles,  then  check  your  estimate  by  use  of  a  protractor. 


2.  Without  using  a  protractor  draw  several  angles  of  as 
near  45°  as  you  can,  and  in  different  positions.  Check  the  ac- 
curacy of  your  drawings  by  measuring  the  angles  with  a 
protractor.  Similarly,  draw  angles  of  10°,  80°,  75°,  15°,  50°, 
and  30°.  Check  the  accuracy  of  your  drawing  in  each  case. 

You  will  find  that  practice  will  enable  you  to  estimate  lengths 
and  the  size  of  angles  with  a  fair  degree  of  accuracy. 


232  GEOMETRY  OF  POSITION 

3.  Indicate  on  paper  a  direction  30°  west  of  south  of  the 
school ;  20°  east  of  north ;  20°  east  of  south. 

By  30°  west  of  south  is  meant  a  direction  making  an  angle  of  30° 
with  a  line  running  directly  south,  and  to  the  west  of  that  line. 

4.  It  is  found  that  a  submarine  cable  is  broken  5  mi.  from 
a  certain  lighthouse  and  30°  west  of  south.    Draw  a  plan, 
using  the  scale  of  1  in.  to  1  mi.,  and  show  where  the  repair 
ship  must  grapple  to  bring  up  the  ends  for  splicing. 

5.  A  boy  starts  to  walk  on  a  straight  road  which  runs 
20°  east  of  north.    After  he  has  gone  3  mi.  he  turns  on  an- 
other straight  road  and  walks  2  mi.  due  west.    Is  he  then 
east  or  west  of  his  starting  point,  and  how  many  degrees  ? 

6.  A  flagstaff   CD  stands  on  the  top  of  a  mound  the 
height  BC  of  which  is  known  to  be  30  ft. 

From  the  point  A  the  angle  BAD  is  ob- 
served to  be  50°  and  the  angle  CAD  to  be 
20°.  Draw  a  diagram  to  scale  and  find  ap- 
proximately the  height  of  the  flagstaff  CD.  <•_ B 

7.  Wishing  to  find  the  length  of  a  pond,  some  boys  staked 
out  a  right  triangle  as  shown  in  the  figure.    Using  a  pro- 
tractor they  found  that  the  angle  A  was  40°, 

and  they  measured  AB,  finding  it  to  be  500  ft. 
Draw  the  figure  to  the  scale  of  1  in.  to  200  ft. 
and  thus  find  the  length  of  the  pond. 

8.  Some  men  buried  a  chest  150  yd.  from  a  tree  A  and 
250  yd.  from  another  tree  B,  which  was  200  yd.  due  north 
of  A.    On  returning  some  years  later  they  found  that  the 
tree  B  had  disappeared.    Draw  a  plan  to  scale  and  show 
how  they  can  find  the  buried  chest,  provided  that  they 
have  a  compass  and  remember  on  which  side  of  AB  they 
hid  the  chest. 


USES  OF  ANGLES  233 

9.  A  seaport  is  on  a  straight  coast  which  runs  due  north 
and  south.  A  steamer  sails  from  it  in  a  direction  30°  west 
of  north  at  the  rate  of  15  mi.  an  hour.  When  will  she  be 
25  mi.  from  the  coast,  and  how  far  will  she  be  from  the 
seaport  at  that  time  ? 

10.  Some  boys  wished  to  determine  the  height  of  a  cliff. 
They  found  the  distance  AC  to  be  90  ft.  and 

the  angle  BAC  to  be  45°.  They  then  drew  the 
figure  to  scale  and  determined  the  height  of 
the  cliff.  What  is  the  height  of  the  cliff  ? 

11.  While  a  ship  is  steaming  due  east  at  the  rate  of 
20  knots  an  hour,  the  lookout  observes  a  light.    At  9  P.M. 
the  light  is  due  north,  at  9.15  P.M.  it  is  10°  west  of  north, 
and  at  9.30  P.M.  it  is  20°  west  of  north.    Determine  by  a 
drawing  whether  the  light  observed  is  stationary. 

A  knot  is  generally  taken  as  6080  ft.,  this  being  approximately 
the  sea  mile ;  the  statute  mile  used  on  land  is  5280  ft. 

12.  There  is  a  seaport  A  on  a  straight  coast  running  east 
and  west.    A  rock  B  lies  3000  yd.  from  A  and  30°  north  of 
east  from  the  coast  line.    A  ship  steams  from  A  in  a  direction 
20°  north  of  east  from  the  coast  line.   What  is  the  nearest 
approach  of  the  ship  to  the  rock?    How  far  is  the  ship 
from  the  coast  at  that  time  ? 

13.  A  man  knows  a  tree  across  a  stream  to  be  60  ft.  high ; 
he  finds  the  angle  at  his  eye  as  shown  in  the  figure  to  be  30°, 
and  he  knows  that  his  eye  is  5  ft.  from  the 

ground.     Make  a  drawing  to  scale  and  de- 
termine the  width  of  the  stream. 

14.  Two    towers   of  equal   height   are  " 
300  ft.  apart.    From  the  foot  of  each  tower 

the  top  of  the  other  tower  makes  an  angle  of  30°  with  a 
horizontal  line.    Determine  the  height  of  the  towers. 


234  GEOMETRY  OF  POSITION 

Exercise  59.    Miscellaneous  Problems 

1.  A  ship  steams  from  a  seaport  A  in  a  direction  50* 
east  of  north.   A  dangerous  rock  lies  northeast  from  A  and 
5000  yd.  from  the  coast  running  east  and  west  through  A. 
Find  how  near  the  ship  approaches  to  the  rock. 

2.  Two  sides  of  a  rectangular  field  ABCD  are  80  rd. 
and  95  rd.    A  tree  in  the  field  stands  120  ft.  from  AB  and 
160  ft.  from  BC.    Draw  the  figure  to  scale  and  show  the 
position  of  the  tree. 

3.  A  man  computes  that  he  should  buy  16  T.  of  soft 
coal  for  his  winter  supply.    His  cellar  is  so  arranged  that 
he  can  make  a  coal  bin  18  ft.  long,  and  the  height  of  the 
cellar  is  8^  ft.    How  wide  a  bin  should  be  constructed  if 
the  top  of  the  coal  is  to  be  a  foot  from  the  ceiling  when 
the  bin  contains  16  T.  of  coal  ? 

Allow  35  cu.  ft.  of  coal  to  the  ton. 

4.  A  boy  standing  due    south   of   a   flagpole   finds   its 
angle  of  elevation,  that  is,  the  angle  to  the  top,  to  be  20°. 
After  he  has  walked  280  ft.  to  the  northwest   on  level 
ground,  he  sees  the  flagpole  to  the  northeast.     Draw  the 
figure  to  scale  and  determine  the  heig  t  of  the  flagpole. 

5.  How  long  a  shadow  will  the  flagpole  mentioned  in 
Ex.  4  cast  when  the  sun's  angle  of  elevation  is  40°  ? 

6.  A  tree  90  ft.  high  casts  a  shadow  140  ft.  long.    Draw 
the  figure  to  scale  and  find  the  sun's  angle  of  elevation. 

7.  A  military  commander  standing  1000  ft. -from  a  fort 
finds  its  angle  of  elevation  to  be  30°.    Draw  the  figure  to 
scale  and  determine  the  height  of  the  fort. 

8.  In  order  to  check  his  work  in  Ex.  7  the  commander 
took  the  angle   of  elevation  from  a  point  1200  ft.  from 
the  fort.    What  did  he  find  this  angle  to  be  ? 


OUTDOOR  WORK  235 

Exercise  60.    Optional  Outdoor  Work 

1.  Determine  the  height  of  a  tree  on  or  near  your  school 
grounds  by  finding  the  angle  of  elevation  of  the  top  from 
two  different  positions  and  measuring  the  distance  of  each 
position  from  the  tree. 

2.  Two  boys  wishing  to  determine  which  of  two  smoke- 
stacks is  the  taller  computed  the  height  of  each  stack  by 
three  different  methods  and  then  took  the  average  of  the 
results  as  the  correct  height.    What  three  methods  might 
they  have   used  ?    Use  three   methods  to   determine   the 
height  of  some  high  object  near  your  school. 

3.  A  graduating  class   decides  to  present  a  drinking- 
fountain  to  the  school.    The  fountain  is  to  be  placed  30  ft. 
from  a  straight  street  which  runs  in  front  of  the  school 
grounds  and  15  ft.  from  the  front  door  of  the  school  build- 
ing.   How  could  the  members  of  the  class  determine  the 
desired  position  ?    Locate,  if  possible,  such  a  position  on 
your  school  grounds  or  on  some  lot  in  the  vicinity. 

4.  Suggest   two  methods   of   determining  the   distance 
between  two  points  when  the  distance  cannot  be  measured 
directly.    Use  both  methods  to  determine  the  distance  be- 
tween two  easily  accessible  points  on  your  school  ground. 
Check  the  accuracy  of  each  method  by  actually  measuring 
the  required  distance. 

5.  A  hawk's  nest  is  observed  in  a  high  tree  and  some 
distance  below  the  top  of  the  tree.    Suggest  two  methods 
by  which  the  height  of  the  nest  may  be  determined  other 
than  by  direct  measurement.    Check  the  accuracy  of  the 
two  methods  by  applying  them  to  a  similar  situation  where 
the  required  distance  can  be  actually  measured  as  a  check 
upon  the  accuracy  of  the  methods. 


JMl 


236  GEOMETRY  OF  POSITION 

Exercise  61.   Problems  without  Figures 

1.  A  workman  has  a  circular  disk  of  metal  and  wishes 
to  find  its  exact  center.    How  should  he  proceed? 

2.  A  man  wishes  to  set  out  a  tree  so  that  it  shall  be 
equally  distant  from  three  trees  which  are  not  in  the  same 
straight  line.   How  should  he  proceed  to  find  the  position  ? 

3.  A  contractor  wishes  to  tap  a  water  main  at  a  point 
equidistant  from  two  hydrants.    How  should  he  proceed 
to  find  the  required  point? 

4.  A  city  engineer  is  asked  to  place  an  electric-light  pole 
at  a  point  equidistant  from  two  intersecting  streets  and  at  a 
given  distance  from  the  corner.    How  does  he  do  it  ? 

5.  A  man  wishes  to  build  on  a  corner  lot  a  house  at  a 
given  distance  from  one  street  and  at  another  given  distance 
from  the  other  street.    How  does  he  lay  out  the  plan  ? 

6.  A  drinking-fountain  is  to  be  placed  in  a  park  at  a 
given  distance  from  a  hydrant  which  is  at  the  side  of  the 
street  in  front  of  the  park,  and  at  a  given  shorter  distance 
back  from  the  street.     Show  that  there  are  two  possible 
points,  and  show  how  to  find  them. 

7.  If  you  know  that  two  water  mains  join  somewhere 
under  a  certain  road,  but  you  do  not  know  where,  what 
measurements  could  be  given  you  with  respect  to  one  or 
more  trees  along  the  side  of  the  road  that  would  tell  you 
where  to  dig  to  find  the  point? 

8.  A  straight  electric-light  wire  runs  under  the  floor  of 
a  room.     The  distances  of  one  point  of  the  wire  from  the 
northeast  corner  and  from  the  north  wall  are  known,  and 
also   the  distances   of  another  point  from  the  southwest 
corner  and  from  the  south  wall.     Show  how  to  mark  on 
the  floor  the  course  of  the  wire. 


SUPPLEMENTARY  WORK 


237 


IV.   SUPPLEMENTARY  WORK 

Squares  and  Square  Roots.    If  a  square  has  a  side  4 
units  long,  it  has  an  area  of  16  square  units.    Therefore 
16  is  called  the  square  of  4,  and  4  is  called  the 
square  root  of  16. 

Square  Roots  of  Areas.  Therefore,  consider- 
ing only  the  abstract  numbers  which  represent 
the  sides  and  area, 

The  side  of  a  square  is  equal  to  the  square  root  of  the  area. 

Symbols.  The  square  of  4  is  written  42,  and  the  square 
root  of  16  is  written  Vl6. 

Perfect  Squares.  Such  a  number  as  16  is  called  a  per- 
fect square,  but  10  is  not  a  perfect  square.  We  may  say, 
however,  that  VlO  is  equal  to  3.16  -f ,  because  3.162  is 
very  nearly  equal  to  10. 

Square  Roots  of  Perfect  Squares.  Square  roots  of  perfect 
squares  may  often  be  found  by  simply  fac- 
toring the  numbers. 

For  example,  V441  =  V3  X  3  x  7  X  7 
=  V3  x  7x  3  x.7 
=  V21  x21=21. 

That  is,  we   separate   441   into   its   factors, 

and  then   separate   these   factors   into   two 

equal  groups,  3x7  and  3x7.    Hence  we  see  that  3x7, 

or  21,  is  the  square  root  of  441. 

We  prove  this  by  seeing  that  21  x  21  =  441. 

To  find  the  square  root  of  a  perfect  square,  separate  it 
into  two  equal  factors. 

The  work  in  square  root  may  be  omitted  if  there  is  not  time  for  it. 


238 


SUPPLEMENTARY  WORK 


Square  of  the  Sum  of  Two  Numbers.    Since  47=  40  +  7, 
the  square  of  47  may  be  obtained  as  follows : 

40  +  7 
40  +  7 


402 


402  +  2  x  (40  x  7)  +  72 
=  1600  +  2x280  +49 
=  1600  +  560  +  49 

=  2209. 


+  u 


This  relationship  is  conveniently  seen  in  the  above  figure, 
in  which  the  side  of  the  square  is  40  +  7. 

Every  number  consisting  of  two  or  more  figures  may  be 
regarded  as  composed  of  tens  and  units.  Therefore 

The  square  of  a  number  contains  the  square  of  the  tens, 
plus  twice  the  product  of  the  tens  and  units,  plus  the  square 
of  the  units. 

This  important  principle  in  square  root  should  be  clearly  under- 
stood, both  from  the  multiplication  and  from  the  illustration. 

Separating  into  Periods.  The  first  step  in  extracting  the 
square  root  of  a  number  is  to  separate  the  figures  of  the 
number  into  groups  of  two  figures  each,  called  periods. 

Show  the  class  that  1  =  I2,  100  =  102,  10,000  =  1002,  and  so  it  is 
evident  that  the  square  root  of  any  number  between  1  and  100  lies 
between  1  and  10,  and  the  square  root  of  any  number  between  100 
and  10,000  lies  between  10  and  100.  In  other  words,  the  square  root 
of  any  integral  number  expressed  by  one  figure  or  two  figures  is  a 
number  of  one  figure ;  the  square  root  of  any  integral  number  ex- 
pressed by  three  or  four  figures  is  a  number  of  two  figures  ;  and  so  on. 

Therefore,  if  an  integral  number  is  separated  into  periods  of  two 
figures  each,  from  the  right  to  the  left,  the  number  of  figures  in  the 
square  root  is  equal  to  the  number  of  the  periods  of  figures.  Th* 
last  period  at  the  left  may  have  one  figure  or  two  figures. 


SQUAEE  EOOT  239 

Extracting  the  Square  Root.  The  process  of  extract- 
ing the  square  root  of  a  number  will  now  be  considered, 
although  in  practice  such  roots  are  usually  found  by  tables. 

For  example,  required  the  square  root  of  2209. 

Show  the  class  that  if  we  separate  the  figures  of  the  number  into 
periods  of  two  figures  each,  beginning  at  the  right,  we  see  that  there 
will  be  two  integral  places  in  the  square  root  of  the  number. 

The  first  period,  22,  contains  the  square  of  the  tens'  number  of 
the  root.  Since  the  greatest  square  in  22  is  16,  then  4,  the  square 
root  of  16,  is  the  tens'  figure  of  the  root. 

Subtracting  the  square  of  the  tens,  the  re- 


2209(47 
16 


80 
87 


609 
609 


mainder  contains  twice  the  tens  x  the  units, 
plus  the  square  of  the  units.  If  we  divide  by 
twice  the  tens  (that  is,  by  80,  which  is  2  x  4 
tens),  we  shall  find  approximately  the  units. 
Dividing  609  by  80  (or  60  by  8),  we  have  7 
as  the  units'  figure. 

Since  twice  the  tens  x  the  units,  plus  the 
square  of  the  units,  is  equal  to  (twice  the  tens  +  the  units)  x  the 
units,  that  is,  since  2  x  40  x  7  +  72  =  (2  x  40  +  7)  x  7,  we  add  7  to 
80  and  multiply  the  sum  by  7.  The  product  is  609,  thus  completing 
the  square  of  47.  Checking  the  T/ork,  472  =  2209. 


Exercise  62.    Square  Root 

Find  the  square  roots  of  the  following  numbers : 
1.  3249.  2.  3721.  3.  3969.  4.  5041. 

Find  the  sides  of  squares,  given  the  following  areas : 

5.  6724  sq.  ft.     7.  9025  sq.  ft.     9.  7921  sq.  yd. 

6.  7569  sq.  ft.    8.  9409  sq.  ft.    10.  6889  sq.  ft. 

Find  the  square  roots  of  the  following  fractions  l>y  taking 
the  square  root  of  each  term  of  each  fraction : 


240 


SUPPLEMENTARY  WORK 


Square  Root  with  Decimals.   Find  the  value  of  V151.29. 

Show  the  class  that  the  greatest  square  of  the  tens  in  151.29  is 
100,  and  that  the  square  root  of  100  is  10. 

Then  51.29  contains  2  x  10  x  the  units'  number  of  the  root,  plus 
the  square  of  the  units'  number.  Ask  why  this  is  the  case. 

Dividing  51  by  2  x  10,  or  20,  we  find  that  the  next  figure  of  the 
root  is  2. 

We  have  now  found  12,  the  square  being  100  +  44  =  144. 

Then  7.29  contains  2  x  12  x  the  tenths'  number  of  the  root,  plus 
the  square  of  the  tenths'  number,  because  we  have  subtracted  144, 
which  is  the  square  of  12. 

Dividing  by  24,  we  find  that  the 
tenths'  figure  of  the  root  is  3. 

Hence  the  square  root  of  151.29 
is  12.3. 

If  the  number  is  not  a  perfect 
square,  we  may  annex  pairs  of  zeros 
at  the  right  of  the  decimal  point  and 
find  the  root  to  as  many  decimal  places 
as  we  choose. 


151. 
1 

29(12.3 

20 

22 

51 
44 

240 
243 

7 
7 

29 
29 

Summary  of  Square  Root.  We  now  see  that  the  following 
are  the  steps  to  be  taken  in  extracting  square  root : 

Separate  the  number  into  periods  of  two  figures  each,  be- 
ginning at  the  decimal  point. 

Find  the  greatest  square  in  the  left-hand  period  and  write 
its  root  for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  the  result  from  the  left-hand 
period,  and  to  the  remainder  annex  the  next  period  for  a 
dividend. 

Divide  the  new  dividend  thus  obtained  by  twice  the  part 
of  the  root  already  found.  Annex  to  this  divisor  the  figure 
thus  found  and  multiply  by  the  number  of  this  figure. 

Subtract  this  result,  bring  down  the  next  period,  and  pro- 
ceed as  before  until  all  the  periods  have  been  thus  annexed. 

The  result  is  the  square  root  required. 


SQUARE  ROOT 


241 


9.  63,001. 

10.  21,224,449. 

11.  49,112,064. 

12.  96,275,344. 


Exercise  63.    Square  Root 

Find  the  square  roots  of  the  following : 

1.  12,321.  5.  19.4481. 

2.  54,756.  6.  0.2809. 

3.  110.25.  7.  1176.49. 

4.  8046.09.  8.  82.2649. 

In  Exs.  13-17  give  the  square  roots  to  two  decimal  places  only. 

13.  2.  14.  5.  15.   7.  16.  8.  17.  11. 

18.  Find,  to  the  nearest  hundredth  of  an  inch,  the  side 
of  a  square  whose  area  is  3  sq.  in. 

Square  on  the  Hypotenuse.  As  we  learned  on  page  112, 
in  a  right  triangle  the  side  opposite  the  right  angle  is  called 
the  hypotenuse. 

If  a  floor  is  made  up  of  triangu- 
lar tiles  like  this,  it  is  easy  to  mark 
out  a  right  triangle.  In  the  figure 
it  is  seen  that  the  square  on  the 
hypotenuse  contains  eight  small  tri- 
angles, while  the  square  on  each 
side  contains  four  such  triangles. 
Hence  we  see  that 

The  square  on  the  hypotenuse  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides. 

This  remarkable  fact  is  proved  in  geometry  for  all  right  triangles. 

Given  that  AB  =  12  and  AC=  9,  find  BC. 

Since  JBC2  =  AB2  +  ~AC2,  G 

therefore  lJC2  =  122  +  92, 

or  ~BCZ  =  144  +  81  =  225, 

and  BC  =  V225  =  15.  A.        12 


242  SUPPLEMENTARY  WORK 

Exercise  64.   Square  Root 

1.  How  long  is  the  diagonal  of  a  floor  48  ft.  by  75  ft.  ? 
On  this  page  state  results  to  two  decimal  places  only. 

2.  Find  the   length  of  the  diagonal  of  a  square  that 
contains  9  sq.  ft. 

3.  The  two  sides  of  a  right  triangle  are  40  in.  and  60  in. 
Find  the  length  of  the  hypotenuse. 

4.  What  is  the  length  of  a  wire  drawn  taut  from  the 
top  of  a  75-foot  building  to  a  spot  40  ft.  from  the  foot  ? 

5.  A  telegraph  pole  is  set  perpendicular  to  the  ground, 
and  a  taut  wire,  fastened  to  it  20  ft.  above  the  ground, 
leads  to  a  stake  15  ft.  6  in.  from  the  foot  of  the  pole,  so 
as  to  hold  it  in  place.    How  long  is  the  wire  ? 

6.  A  derrick  for  hoisting  coal  has  its 
arm  27  ft.  6  in.  long.    It  swings  over  an 
opening  22  ft.  from  the  base  of  the  arm. 

How  far  is  the  top  of  the  derrick  above    /      52 
the  opening? 

Reversing  the  procedure  on  page  241,  the  square  on  either  side  is 
equal  to  the  difference  between  what  two  squares  ? 

7.  The  foot  of  a  45-foot  ladder  is  27  ft.  from  the  wall 
of  a  building  against  which  the  top  rests.   How  high  does 
the  ladder  reach  on  the  wall  ?  _B 

8.  To  find  the   length  of  this  pond  a 
class  laid  off  the   right  triangle  ACE  as 
shown.     It  was  found  that  .4(7=428  ft., 

BC  =  321  ft.,  and  AD  =  75  ft.    Find  DB.   /—          ? 

9.  How  far  from  the  wall  of  a  house  must  the  foot  of 
a  36-foot  ladder  be  placed  so  that  the  top  may  touch  a 
window  sill  32  ft.  from  the  ground  ? 


,  .    PYRAMIDS 


243 


Prism.    A  solid  in  which  the  bases  are  equal  polygons 
and  the  other  faces  are  rectangles  is  called  a  prism, 

Volume  of  a  Prism.  It  is  evident  that  we 
may  find  the  volume  of  a  prism  in  the  same 
way  that  we  found  the  volume  of  a  cylinder 
(page  $00).  That  is, 

The  volume  of  a  prism  is  equal  to  the  product  of  the  base 
and  height. 

Pyramid.  A  solid  of  this  shape  in  which  the  base  is 
any  polygon  and  the  other  faces  are  triangles 
meeting  at  a  point  is  called  a  pyramid.  The 
point  at  which  the  triangular  faces  meet  is 
called  the  vertex  of  the  pyramid,  and  the 
distance  from  the  vertex  to  the  base  is  called 
the  altitude  of  the  pyramid.  The  faces  not 
including  the  base  are  called  lateral  faces. 

Volume  of  a  Pyramid.  If  we  fill  a  hollow  prism  with 
water  and  then  pour  the  water  into  a 
hollow  pyramid  of  the  same  base  and 
the  same  height,  as  here  shown,  it  will 
be  found  that  the  pyramid  has  been 
filled  exactly  three  times  with  the  water 
that  filled  the  prism.  Therefore, 


Ttie  volume  of  a  pyramid  is  equal  to  one  third  the  product 
of  the  base  and  height. 

Lateral  Surface  of  a  Pyramid.  The  height  of  a  lateral 
face  of  a  pyramid  is  called  the  slant  height  of  the  pyramid. 

Since  the  area  of  each  lateral  face  of  a  pyramid  is  equal 
to  half  the  product  of  the  base  and  altitude, 

The  area  of  the  lateral  surface  of  a  pyramid  is  equal  to 
the  perimeter  of  the  base  multiplied  by  half  the  slant  height. 


244  SUPPLEMENTARY  WORK 

Lateral  Surface  of  a  Cone.  If  we  should  slit  the  surface 
of  a  cone  and  flatten  it  out,  we  would  have  part  of  a  circle. 

The  terms  "lateral  surface "  and  "slant  height"  will 
be  understood  from  the  study  of  the  pyramid. 

From  our  study  of  the  circle  we  infer  that 

The  lateral  surface  of  a  cone  is  equal  to  the 
circumference  of  the  base  multiplied  by  half  the 
slant  height. 

Volume  of  a  Cone.  In  the  way  that  we  found  the  volume 
of  a  pyramid  we  may  find  the  volume  of  a  cone.  Then 

The  volume  of  a  cone  is  equal  to  one  third  the  product  of 
the  base  and  height. 

Find  the  volume  of  a  cone  of  height  5  in.  and  radius  2  in. 

Area  of  base  is  -^  x  4  sq.  in. 

Volume  is  £  x  5  x  '^?  x  4  cu.  in.  =  20.95  cu.  in. 

This  method  of  calculation  gives  20.95  55T  cu.  in.  as  the  volume,  but 
20.95  cu.  in.  is  a  much  more  practical  form  for  the  answer. 

Teachers  will  observe  that  only  the  simplest  forms  of  prisms, 
pyramids,  and  cones  have  been  considered  in  this  book. 

Exercise  65.    Lateral  Surfaces  and  Volumes 

Find  the  volumes  of  prisms  and  also  the  volumes  of  pyra- 
mids with  the  following  bases  and  heights : 

1.  36sq.in.,  7in.     2.  48  sq.in.,5|dn.     3.  5. 7 sq. in., 4.8  in. 

Find  the  lateral  surfaces  of  pyramids  with  the  following 
perimeters  of  bases  and  slant  heights : 

4.  30  in.,  18  in.       5.  3  ft.  3  in.,  8  in.      6.  5  ft.  9  in.,  10  in. 

Find  the  volumes  of  cones  with  the  following  radii  of  bases 
and  heights : 

7.  14  in.,  6  in.         8.  5.6  in.,  15  in.        9.  49  in.,  15  in. 


CONES  AND  SPHERES  245 

Surface  of  a  Sphere.  If  we.  wind  half  of  the  surface  of 
a  sphere  with  a  cord  as  here  shown,  and  then  wind  with 
exactly  the  same  length  of  the  cord  the  surface  of  a 


cylinder  whose  radius  is  equal  to  the  radius  of  the  sphere 
and  whose  height  is  equal  to  the  diameter,  we  find  that 
the  cord  covers  half  the  curved  surface  of  the  cylinder. 

Therefore  the  surface  of  a  sphere  is  equal  to  the  curved 
surface  of  a  cylinder  of  the  same  radius  and  height. 

We  can  now  easily  show  that 

surface  of  sphere  =  ^-  X  2  x  radius  x  2  x  radius. 

Hence  the  surface  of  a  sphere  is  equal  to  4  times  -j-  times 
the  square  of  the  radius. 

Volume  of  a  Sphere.    It  is  shown  in  geometry  that 

The  volume  of  a  sphere  is  equal  to  ~  times  %j-  times  the 
cube  of  the  radius. 

The  cube  of  r  means  r  x  r  x  r  and  is  written  r8. 

1.  Considering  the  earth  as  a  sphere  of  4000  mi.  radius, 
find  the  surface. 

4  x  3j£  x  40002  =  4  x  3,?  x  16,000,000  =  201,142,857}. 
Therefore  the  surface  is  about  201,143,000  sq.  mi. 

2.  If  a  ball  is  4  ft.  in  diameter,  find  the  volume. 

The  radius  is  £  of  4  ft.,  or  2  ft. 

The  volume  is  f  x  ^  x  2  x  2  x  2  cu.  ft.,  or  33.52  cu.  ft. 


246  SUPPLEMENTARY  WORK 

Exercise  66.    Surfaces  and  Volumes 

1.  If  a  ball  has  a  radius  of  1^  in.,  find  the  surface. 

2.  Find  the  surface  of  a  tennis  ball  of  diameter  2$  in. 

3.  If  a  cubic  foot  of  granite  weighs  165  lb.,  find  the 
weight  of  a  sphere  of  granite  4  ft.  in  diameter. 

4.  A  bowl  is  in  the  form  of  a  hemisphere  4.9  in.  in 
diameter.    How  many  cubic  inches  does  it  contain  ? 

5.  A  ball  4'  6"  in  diameter  for  the  top  of  a  tower  is 
to  be  gilded.    How  many  square  inches  are  to  be  gilded  ? 

6.  A  pyramid  has  a  lateral  surface  of  400  sq.  in.    The 
slant  height  is  16  in.    Find  the  perimeter  of  the  base. 

7.  A  conic  spire  has  a  slant  height  of  34  ft.  and  the 
circumference  of  the  base  is  30  ft.  Find  the  lateral  surface. 

8.  What  is  the  entire  surface  of  a  cone  whose  slant 
height  is  6  ft.  and  the  diameter  of  whose  base  is  6  ft.  ? 

9.  What  is  the  weight  of  a  sphere  of  marble  3  ft.  in 
circumference,  marble  being  2.7  times  as  heavy  as  water 
and  1  cu.  ft.  of  water  weighing  1000  oz.  ? 

10.  Taking  the  earth  as  an  exact  sphere  with  radius 
4000  mi.,  find  the  volume  to  the  nearest  1000  cu.  mi. 

11.  If  1  cu.  ft.  of  a  certain  quality  of  marble  weighs 
173  lb.,  what  is  the  weight  of  a  cylindric  marble  column 
that  is  12ft.  high  and  18  in.  in  diameter? 

12.  How  many  cubic  yards  of  earth  must  be  removed 
in  digging  a  canal  8  mi.  900  ft.  long,  180  ft.  wide,  and 
18ft.  deep? 

13.  A    marble    1^  in.    in   diameter   is   dropped   into    a 
cylindric  jar  5  in.  high  and  4  in.  in  diameter,  half  full  of 
water.   How  much  does  the  marble  cause  the  water  to  rise? 


TABLES  FOR  REFERENCE 

LENGTH 

12  inches  (in.)  =1  foot  (ft.) 

3  feet  =  1  yard  (yd.) 
5|  yards,  or  16§  feet  =  1  rod  (rd.) 
320  rods,  or  5280  feet  =  1  mile  (mi.) 

SQUARE  MEASURE 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  square  feet  =  1  square  yard  (sq.  yd.) 
30|  square  yards  =  1  square  rod  (sq.  rd.) 
160  square  rods  =  1  acre  (A.) 

640  acres  =  1  square  mile  (sq.  mi.) 

CUBIC  MEASURE 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.) 
128  cubic  feet  =  1  cord  (cd.) 

WEIGHT 

16  ounces  (oz.)  =  1  pound  (Ib.) 
2000  pounds  =  1  ton  (T.) 

LIQUID  MEASURE 

4gills(gi.)=l  pint(pt.) 
2  pints  =  1  quart  (qt.) 

4  quarts  =  1  gallon  (gal.) 
31|  gallons  =1  barrel  (bbl.) 

2  barrels  =1  hogshead  (hhd.) 

247 


248  TABLES  FOE  REFERENCE 


DRY  MEASURE 

2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 

TIME 

60  seconds  (sec.)  —  1  minute  (min.) 
60  minutes  =  1  hour  (hr.) 
24  hours  =  1  day  (da.) 

7  days  =  1  week  (wk.) 
12  months  (mo.)  =1  year  (yr.) 

365  days  =  1  common  year 

366  days  =  1  leap  year 

VALUE 

10  mills  =  1  cent  (£  or  ct.) 
10  cents  =  1  dime  (d.) 
10  dimes  =  1  dollar  ($) 

ANGLES  AND  ARCS 

60  seconds  (60")  =1  minute  (I') 
60  minutes  =  1  degree  (1°) 

COUNTING 

12  units  =  1  dozen  (doz.) 
12  dozen,  or  144  units  =l-gross  (gr.) 
12  gross,  or  1728  units  =1  great  gross 

PAPER 

24  sheets  =  1  quire 
500  sheets  =  1  ream 

Formerly  480  sheets  of  paper  were  called  a  ream.  The  word  "quire  " 
is  now  used  only  for  folded  note  paper,  other  paper  being  usually 
sold  by  the  pound. 


INDEX 


PAGE 

Account  ....      2,  4,  23,  57,  87 
Accurate  proportions     .    .    .    140 

Acute  angle 112 

triangle 112 

Addition 33 

Aliquot  parts 42 

Altitude 150,  200,  243 

Amount  of  a  note 94 

Angle 112 

Arc 115 

Area  162,164,167,169,172,174,196 

Balance 2 

Bank 79,  86,  90,  97 

Base 114,  150,  200 

Bill .      48 

Bisection 124 

Cash  check 43 

Center 149,  150 

Check 2,  43,  92,  155 

Circle  ....    115,  149,  194,  196 
Circumference    .    .     115,  149,  194 

Commercial  paper 99 

Compound  interest     ....      86 

Cone 150,  244 

Congruent  figures 115 

Constructing  triangles    .    .    .    116 

Creditor 49 

Cube 198 

Cylinder 150,  200 

Debtor 49 

Deposit  slip 90 


PAGE 

Diameter    .    .     115,  149,  150,  194 

Discount 44,  46,  97 

Distances 222 

Dividing  a  line 130 

Drawing  instruments      .    .    .    115 
to  scale 136,  204 

Ellipse 149 

Equilateral  triangle    .    .    112,  118 

Face  of  a  note 94 

Formulas     164,167,170,172,194, 

196,  200,  201 

Fractions 67 

Geometric  figures 112 

measurement 155 

patterns 132 

Height 150,200 

Hypotenuse 112,  241 

Indorsement 92,  95 

Instruments,  drawing    .    .    .    115 

Interest 81 

Invoice 50 

Isosceles  triangle    .    .    .    112,  118 

Lateral  surface  ....    243,  244 

Lathing 203 

Length 165 

Line 114 

List  price 44 

Locating  points 219 


249 


250 


INDEX 


FACE 

Maker  of  a  note 96 

Map  drawing 217 

Marked  price 44 

Material  for  daily  drill  .    .    .    105 

Metric  measures 205 

Miscellaneous  problems .    .  30,  73, 

102,  212,  234 

Multiplication 37 

Net  price 44 

Note 94 

Oblique  angle 112 

Obtuse  angle 112 

triangle 112 

Outdoor  work       153,  156,  178,  192 

213,  235 
Overhead  charges 74 

Pantograph 145 

Parallel  lines 128 

Parallelogram     ....     114,  167 

Payee 92,  95 

Per  cent 6,  7 

Percentage  problems  ....      17 

Perimeter 112,  114 

Perpendicular     ....    121,  222 

Photograph 143 

Plane  figures 149 

Plastering 203 

Polygon 114,  174 

Position  .    .    .      215,  219,  227,  229 
Postal  savings  bank    ....      89 

Price  list 24,  47 

Principal 81,  94 

Problems  without  figures   .    .    154 

214,  236 
without  numbers      32,  54,  66, 

76,  104 
Proceeds  97 


PACK 

Promissory  note 94 

Proportion 184 

Proportional 186 

Protractor 115 

Pyramid 243 

Quadrilateral 114 

Radius     .    .    .     115,  149,  150,  194 

Rate  of  interest 81,  95 

Ratio 145,  182 

Receipted  bill 49 

Rectangle 114,  164 

Rectangular  solid 198 

Review  drill  31,  53,  65,  75,  103,  152 

Right  angle 112 

triangle 112 

Savings  bank      79,  86 

Several  discounts 46 

Short  cuts  in  multiplication  .  38 
Similar  figures  ....  141,  186 
Six  per  cent  method  ....  100 

Size 155 

Sphere 150,  245 

Square 114,  196,  237 

roots 237 

Squared  paper 162 

Subtraction 35 

Symmetry 147 

Tables  for  reference  ....    247 

Trapezoid 114,  172 

Triangle      112,  169 

Units  of  area 164 

Uses  of  angles 231 

Vertex  .  .  .  112,  114,  150,  243 
Volume  .  198,  200,  243,  244,  245 


JAM  3     1939 


Y  3     1949- 
OCT  3  i  'a30 
1  4  1958 


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